Practice Questions for NET JRF Real Analysis Assignment: Differentiability

1. The value of \(\lim _{x \rightarrow 0} \frac{1}{x} \int_{x}^{2 x} e^{-t^{2}} d t\)

(a.) does not exist.
(b.) is infinite.
(c.) exists and equals 1.
(d.) exists and equals 0.

2. Let \(f\) be a bounded function on \(\mathbb{R}\) and \(\begin{array}{lll}a \in \mathbb{R} . \quad \text { For } & \delta>0, \quad \text { let } \\ \omega(a, \delta)=\sup |f(x)-f(a)|, & x \in[a-\delta, a+\delta] .\end{array}\) Then

(a.) \(\omega\left(a, \delta_{1}\right) \leq \omega\left(a, \delta_{2}\right)\) if \(\delta_{1} \leq \delta_{2}\).
(b.) \(\lim _{\delta \rightarrow 0+} \omega(a, \delta)=0\) for all \(a \in \mathbb{R}\).
(c.) \(\lim _{\delta \rightarrow 0+} \omega(a, \delta)\) need not exist.
(d.) \(\lim_{\delta \rightarrow 0^{+}} \omega(a, \delta)=0\) if and only if \(f\) is continuous at \(a\).

3. Let \(f\) be a continuously differentiable function on \(\mathbb{R}\). Suppose that \(L=\lim _{x \rightarrow \infty}\left(f(x)+f^{\prime}(x)\right)\) exists. If \(0

(a.) If \(\lim _{x \rightarrow \infty} f^{\prime}(x)\) exists, then it is 0.
(b.) If \(\lim _{x \rightarrow \infty} f(x)\) exists, it is \(L\).
(c.) If \(\lim _{x \rightarrow \infty} f^{\prime}(x)\) exists, then \(\lim _{x \rightarrow \infty} f(x)=0\).
(d.) If \(\lim _{x \rightarrow \infty} f(x)\) exists, then \(\lim _{x \rightarrow \infty} f^{\prime}(x)=L\).

4. Let \(A \subseteq \mathbb{R}\) and \(f: A \rightarrow \mathbb{R}\) be given by \(f(x)=x^{2}\). Then \(f\) is uniformly continuous if

(a.) \(A\) is a bounded subset of \(\mathbb{R}\)
(b.) \(A\) is a dense subset of \(\mathbb{R}\)
(c.) \(A\) is an unbounded and connected subset of \(\mathbb{R}\)
(d.) \(A\) is an unbounded and open subset of \(\mathbb{R}\)

5. Let \(P(x)=\left(\frac{5}{13}\right)^{x}+\left(\frac{12}{13}\right)^{x}-1\) for all \(x \in \mathbb{R}\). Then which of the following statement(s) is(are) TRUE?

(a.) The equation \(P(x)=0\) has exactly one solution in \(\mathbb{R}\)
(b.) \(P(x)\) is strictly increasing for all \(x \in \mathbb{R}\)
(c.) The equation \(P(x)=0\) has exactly two solutions in \(\mathbb{R}\)
(d.) \(P(x)\) is strictly decreasing for all \(x \in \mathbb{R}\)

6. The function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is given by \(f(x)=e^{|x|+x^{2}}+\left|x^{2}-1\right|\) Which of the following is true about the function \(f\) ?

(a.) It is not differentiable exactly at three points of \(\mathbb{R}\).
(b.) It is not differentiable at \(x=0\).
(c.) It is differentiable at \(x=2\).
(d.) It is not differentiable at \(x=1\) and \(x=-1\).

7. The function \(f(x)=a_{0}+a_{1}|x|+a_{2}|x|^{2}+a_{3}|x|^{3}\) is differentiable at \(x=0\)

(a.) For no values of \(a_{0}, a_{1}, a_{2}, a_{3}\)
(b.) For any value of \(a_{0}, a_{1}, a_{2}, a_{3}\)
(c.) Only if \(a_{1}=0\)
(d.) Only if both \(a_{1}=0\) and \(a_{3}=0\)

8. Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a function that satisfies \(|f(x)-f(y)| \leq|x-y|^{\beta}, \beta>0\). Which of the following is correct?

(a.) If \(\beta=1\) then \(f\) is differentiable
(b.) If \(\beta>0\) then \(f\) is uniformly continuous
(c.) If \(\beta>1\) then \(f\) is a constant function
(d.) \(f\) must be a polynomial

9. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function with period \(\rho>0\). Then \(g(x)=\int_{x}^{x+\rho} f(t) dt\) is a

(a.) Constant function
(b.) Continuous function
(c.) Continuous function but not differentiable
(d.) Neither continuous nor differentiable

10. Let \(A\) be the set of rational numbers in the open interval \((0,7)\) and \(f: A \rightarrow \mathbb{R}\) be a uniformly continuous function. Which of the following are true?

(a.) \(f\) is bounded
(b.) \(f\) is necessarily a constant function
(c.) \(f\) is differentiable on \((0,7)\)
(d.) \(f\) is differentiable at all the rational points in \((0,7)\)

11. Let \(f\) be a twice differentiable function on \(\mathbb{R}\). Given that \(f''(x)>0\) for all \(x \in \mathbb{R}\),

(a.) \(f(x)=0\) has exactly two solutions on \(\mathbb{R}\)
(b.) \(f(x)=0\) has a positive solution if \(f(0)=0\) and \(f'(0)=0\)
(c.) \(f(x)=0\) has no positive solution if \(f(0)=0\) and \(f'(0)>0\)
(d.) \(f(x)=0\) has no positive solution if \(f(0)=0\) and \(f'(0)<0\)

12. Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. Then which of the following statements are necessarily true?

(a.) If \(f'(x) \leq r<1\) for all \(x \in \mathbb{R}\), then \(f\) has at least one fixed point.
(b.) If \(f\) has a unique fixed point, then \(f'(x) \leq r<1\) for all \(x \in \mathbb{R}\)
(c.) If \(f\) has a unique fixed point, then \(f'(x) \geq r>-1\) for all \(x \in \mathbb{R}\)
(d.) None of these

13. Let \(f(x)=\sin x-x+\frac{x^{3}}{3!}\) and \(g(x)=\cos x-1+\frac{x^{2}}{2!}\) for \(x \in \mathbb{R}\). Which of the following statements are correct?

(a.) \(f(x) \geq 0\) for all \(x>0\)
(b.) \(g\) is an increasing function on \([0, \infty)\)
(c.) \(g\) is a decreasing function on \([0, \infty)\)
(d.) \(f\) is a decreasing function on \([0, \infty)\)

14. Let \(I=[0,1] \subset \mathbb{R}\). For \(x \in \mathbb{R}\), let \(\varphi(x)=\operatorname{dist}(x, I)=\inf \{|x-y|: y \in I\}\). Then

(a.) \(\varphi(x)\) is discontinuous somewhere on \(\mathbb{R}\)
(b.) \(\varphi(x)\) is continuous on \(\mathbb{R}\) but not continuously differentiable exactly at \(x=0\)
(c.) \(\varphi(x)\) is continuous on \(\mathbb{R}\) but not continuously differentiable exactly at \(x=0\) and \(x=1\)
(d.) \(\varphi(x)\) is differentiable on \(\mathbb{R}\)

15. Consider the function \(f(x)=\cos (|x-5|)+\sin (|x-3|)+|x+10|^{3}-(|x|+4)^{2}\) At which of the following points is \(f\) not differentiable?

(a.) \(x=5\)
(b.) \(x=3\)
(c.) \(x=-10\)
(d.) \(x=0\)

16. Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \(f(x)=\begin{cases} x^{2} & \text { if } x<0, \\ 2x+x^{2} & \text { if } x \geq 0 \end{cases}\) Then which of the following statements are correct?

(a.) \(f''(x)=2\) for all \(x \in \mathbb{R}\)
(b.) \(f''(0)\) does not exist
(c.) \(f'(x)\) exists for each \(x \neq 0\)
(d.) \(f'(0)\) does not exist

17. Consider the function \(f(x)=|\cos x|+|\sin (2-x)|\) At which of the following points is \(f\) not differentiable?

(a.) \(\left\{(2n+1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}\)
(b.) \(\{n\pi: n \in \mathbb{Z}\}\)
(c.) \(\{n\pi+2: n \in \mathbb{Z}\}\)
(d.) \(\left\{\frac{n\pi}{2}: n \in \mathbb{Z}\right\}\)

18. Let

\(F=\left\{f: \mathbf{R} \rightarrow \mathbf{R}:|f(x)-f(y)| \leq K|(x-y)|^{\alpha}\right\}\)

For all \(x, y \in \mathbf{R}\) and for some \(\alpha>0\) and some \(K>0\). Which of the following is/are true?

(a.) Every \(f \in F\) is continuous
(b.) Every \(f \in F\) is uniformly continuous
(c.) Every differentiable function \(f\) is in \(F\)
(d.) Every \(f \in F\) is differentiable

19. Let \(f(x)=x^{2} \sin \frac{1}{x^{2}}\), and \(f(0)=0\) for \(x \neq 0\)

(a.) \(f\) is differentiable everywhere with continuous derivative
(b.) \(f\) is differentiable everywhere other than the point 0
(c.) \(f\) is nowhere differentiable
(d.) \(f\) is differentiable but its derivative is not continuous

20. Let \(f:(0,1) \rightarrow \mathbb{R}\) be continuous. Suppose that

\(|f(x)-f(y)| \leq |\cos x-\cos y|\) for all \(x, y \in (0,1)\)

Then

(a.) \(f\) is discontinuous at least at one point in \((0,1)\)
(b.) \(f\) is continuous everywhere on \((0,1)\) but not uniformly continuous on \((0,1)\)
(c.) \(f\) is uniformly continuous on \((0,1)\)
(d.) \(\lim _{x \rightarrow 0} f(x)\) exists

21. The function \(f(x)=|x|+3\) is

(a.) Continuous as well as differentiable on \(\mathbb{R}\)
(b.) Continuous on \(\mathbb{R}\) but not differentiable anywhere on \(\mathbb{R}\)
(c.) Continuous on \(\mathbb{R}\) and differentiable at all but finitely many points on \(\mathbb{R}\)
(d.) Non-continuous on \(\mathbb{R}\)

22. Let \(f(x)=|\sin \pi x|\), \(x \in \mathbb{R}\). Then

(a.) \(f\) is continuous nowhere
(b.) \(f\) is continuous everywhere and differentiable nowhere
(c.) \(f\) is continuous everywhere and differentiable everywhere except at integral values of \(x\)
(d.) \(f\) is differentiable everywhere

23. The function \(f(x)=1-|1-x|\) on \(\mathbb{R}\) is

(a.) Not continuous
(b.) Continuous but not differentiable
(c.) Differentiable but not continuous
(d.) Differentiable at only one point

24. \(f:[0,1] \rightarrow \mathbb{R}\) is a function. Which of the following is possible?

(a.) \(f\) is differentiable but not continuous
(b.) \(f\) is unbounded and continuous
(c.) \(f\) is differentiable but its derivative is not continuous
(d.) \(f\) is continuous but not integrable

25. Let \(f:\mathbb{R} \rightarrow \mathbb{R}\) be differentiable. Which of these will follow from the mean value theorem?

(a.) For all \(a, b\) in \(\mathbb{R}\), if \(a
(b.) For some \(a, b\) in \(\mathbb{R}\) and for all \(c\) in \((a, b)\), \(\frac{f(b)-f(a)}{b-a}=f'(c)\)
(c.) For all \(a, b\) in \(\mathbb{R}\), there is some \(c\) in \((a, b)\) such that \(\frac{f(b)-f(a)}{b-a}=f'(c)\)
(d.) For all \(a, b\) in \(\mathbb{R}\), there is a unique element \(c\) in \((a, b)\) such that \(\frac{f(b)-f(a)}{b-a}=f'(c)\)

26. The function \(f(x)=\left\{\begin{array}{cc}x \sin \left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x=0\end{array}\right.\) is

(a.) Differentiable at 0 but not continuous
(b.) Has \(2^{\text {nd }}\) derivative at the origin
(c.) Continuous at the origin but not differentiable
(d.) Neither continuous nor differentiable at the origin

27. The set of points where \(f(x)=|\sin x|\) is not differentiable is

(a.) Empty
(b.) \(\{0\}\)
(c.) \(\{k \pi: k \in \mathbf{Z}\}\)
(d.) \(\left\{\frac{k \pi}{2}: k \in \mathbf{Z}\right\}\)

28. Let \(f:(0,2) \rightarrow \mathbb{R}\) be defined by \(f(x)=\left\{\begin{array}{cc}x^{2} & \text { If } x \text { is rational } \\ 2 x-1 & \text { If } x \text { is irrational }\end{array}\right.\) Then

(a.) \(f\) is differentiable exactly at one point
(b.) \(f\) is differentiable exactly at two points
(c.) \(f\) is not differentiable at any point in \((0,2)\)
(d.) \(f\) is differentiable at every point in \((0,2)\)

29. \(f:\mathbb{R} \rightarrow \mathbb{R}\) is such that \(f(0)=0\) and \(\left|\frac{d f}{d x}(x)\right| \leq 5\) for all \(x\). We can conclude that \(f(1)\) is in

(a.) \((5,6)\)
(b.) \([-5,5]\)
(c.) \((-\infty,-5) \cup (5, \infty)\)
(d.) \([-4,4]\)

30. Let \(f:\mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function such that \(\sup _{x \in \mathbb{R}}\left|f^{\prime}(x)\right| < \infty\). Then

(a.) \(f\) maps a bounded sequence to a bounded sequence.
(b.) \(f\) maps a Cauchy sequence to a Cauchy sequence.
(c.) \(f\) maps a convergent sequence to a convergent sequence.
(d.) \(f\) is uniformly continuous.

31. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a twice continuously differentiable function, with \(f(0)=f(1)=f^{\prime}(0)=0\). Then

(a.) \(f^{n}\) is the zero function.
(b.) \(f^{\prime \prime}(0)\) is zero.
(c.) \(f^{\prime \prime}(x)=0\) for some \(x \in (0,1)\).
(d.) \(f^{\prime \prime}\) never vanishes.

32. Let \(f:[0, \infty) \rightarrow [0, \infty)\) be a continuous function. Which of the following is correct?

(a.) There is \(x_{0} \in [0, \infty)\) such that \(f(x_{0})=x_{0}\)
(b.) If \(f(x) \leq M\) for all \(x \in [0, \infty)\) for some \(M>0\), then there exists \(x_{0} \in [0, \infty)\) such that \(f(x_{0})=x_{0}\).
(c.) If \(f\) has a fixed point, then it must be unique.
(d.) \(f\) does not have a fixed point unless it is differentiable on \((0, \infty)\).

33. Take the closed interval \([0,1]\) and open interval \((1 / 3,2 / 3)\). Let \(K=[0,1] \backslash (1 / 3,2 / 3)\). For \(x \in [0,1]\) define \(f(x)=d(x, K)\) where \(f(x)=d(x, K)=\inf \{\mid x-y \| y \in K\}\). Then

(a.) \(f:[0,1] \rightarrow \mathbb{R}\) is differentiable at all points of \((0,1)\).
(b.) \(f:[0,1] \rightarrow \mathbb{R}\) is not differentiable at \(1 / 3\) and \(2 / 3\).
(c.) \(f:[0,1] \rightarrow \mathbb{R}\) is not differentiable at \(1 / 2\).
(d.) \(f:[0,1] \rightarrow \mathbb{R}\) is not continuous.

34. Let \(X=(0,1) \cup (2,3)\) be an open set in \(\mathbb{R}\). Let \(f\) be a continuous function on \(X\) such that the derivative \(f^{\prime}(x)=0\) for all \(x\). Then the range of \(f\) has

(a.) Uncountable number of points
(b.) Countably infinite number of points
(c.) At most 2 points
(d.) At most 1 point

35. If \(f(x)\) is a real-valued function defined on \([0, \infty]\) such that \(f(0)=0\) and \(f^{\prime \prime}(x)>0\) for all \(x\), then the function \(h(x)=\frac{f(x)}{x}\) is

(a.) Increasing in \([0, \infty]\)
(b.) Decreasing in \([0,1]\)
(c.) Increasing in \([0,1]\) and decreasing in \([1, \infty]\)
(d.) Decreasing in \([0,1]\) and increasing in \([1, \infty]\)

36. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function and \(f(x+1)=f(x)\) for all \(x \in \mathbb{R}\). Then

(a.) \(f\) is bounded above, but not bounded below.
(b.) \(f\) is bounded above and below, but may not attain its bounds.
(c.) \(f\) is bounded above and below and \(f\) attains its bounds.
(d.) \(f\) is uniformly continuous.

37. Decide which of the following functions are uniformly continuous on \((0,1)\).

(a.) \(f(x)=e^{x}\)
(b.) \(f(x)=1/x\)
(c.) \(f(x)=\tan \left(\frac{\pi x}{2}\right)\)
(d.) \(f(x)=\sin (x)\)

38. Which of the following statements is (are) true on the interval \(\left(0, \frac{\pi}{2}\right)\)?

(a.) \(\cos x<\cos (\sin x)\)
(b.) \(\tan x < x\)
(c.) \(\sqrt{1+x}<1+\frac{x}{2}-\frac{x^{2}}{8}\)
(d.) \(\frac{1-x^{2}}{2}<\ln (2+x)\)

39. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function with \(f(0)=0\). If for all \(x \in \mathbb{R}\), \(1

(a.) \(f\) is unbounded
(b.) \(f\) is increasing and bounded
(c.) \(f\) has at least one zero
(d.) \(f\) is periodic

40. Let \(f(x)=\frac{1}{1+|x|}+\frac{1}{1+|x-1|}\) for all \(x \in [-1,1]\). Then which one of the following is TRUE?

(a.) Maximum value of \(f(x)\) is \(\frac{3}{2}\)
(b.) Minimum value of \(f(x)\) is \(\frac{1}{3}\)
(c.) Maximum of \(f(x)\) occurs at \(x=\frac{1}{2}\)
(d.) Minimum of \(f(x)\) occurs at \(x=1\)

41. Let \(f(x)=\left\{\begin{array}{cl}\frac{\sin x}{x} & \text{ if } x \neq 0 \\ 1 & \text{ if } x=0\end{array}\right.\). Then

(a.) \(f\) is not continuous
(b.) \(f\) is continuous but not differentiable
(c.) \(f\) is differentiable
(d.) \(f\) is not bounded