Practice Questions for CSIR NET Real Analysis : Function of Several Variables

1. Let \(A=\left\{(x, y) \in \mathbb{R}^2: x+y \neq-1\right\}\). Define \(f: A \rightarrow \mathbb{R}^2\) by \(f(x, y)=\left(\frac{x}{1+x+y}, \frac{y}{1+x+y}\right)\). Then,

(a.) The Jacobian matrix of \(f\) does not vanish on \(A\)
(b.) \(f\) is infinitely differentiable on \(A\)
(c.) \(f\) is injective on \(A\)
(d.) \(f(A)=\mathbb{R}^2\)

2. Define \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) by \(f(x, y)=\left(x+2y+y^2+|xy|, 2x+y+x^2+|xy|\right)\) for \((x, y) \in \mathbb{R}^2\). Then,

(a.) \(f\) is discontinuous at \((0,0)\)
(b.) \(f\) is continuous at \((0,0)\) but not differentiable at \((0,0)\)
(c.) \(f\) is differentiable at \((0,0)\)
(d.) \(f\) is differentiable at \((0,0)\) and the derivative \(D f(0,0)\) is invertible

3. Let \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be given by \(f(x, y)=(x+y, xy)\). Then,

(a.) \(f\) is not differentiable at the point \((0,0)\)
(b.) The derivative of \(f\) is invertible except on the set \(\left\{(x, y) \in \mathbb{R}^2 \mid x=y\right\}\)
(c.) The inverse image of each point in \(\mathbb{R}^2\) under \(f\) has at most two elements
(d.) \(f\) is surjective

4. Let \(f: \mathbb{R}^n \rightarrow \mathbb{R}^n\) be a differential function. Let \(D f(x)\) be the derivative of \(f\) at \(x \in \mathbb{R}^n\). Which of the following is/are correct?

(a.) \(D f(x)(u)=0\) for all \(u\) in \(\mathbb{R}^n\)
(b.) \(D f(x)(u)=0\) for all \(u\) in \(\mathbb{R}^*\) and some \(\bar{x} \in \mathbb{R}^n\) only if \(f\) is a constant
(c.) \(D f(x)(u)=0\) for all \(u \in \mathbb{R}^n\) and all \(x \in \mathbb{R}^n\) only if \(f\) is a constant
(d.) If \(f\) is not a constant function, then \(D f(x)\) is a one-to-one function for some \(x \in \mathbb{R}^n\)

5. If \(f: S \rightarrow S\) is a function, then we denote by \(f^k\), the function \(f \circ f \circ \cdots \circ f\) (\(k\) times). Let \(f_1\) and \(f_2\) be two functions defined on \(\mathbb{R}^2\) as follows:

\(f_1(x, y) = (x+1, y+3), f_2(x, y) = (x-3, y-2)\). Then

(a.) For any positive integer \(k\), there exists a unique \((a, b) \in \mathbb{R}^2\), such that \(f_1^k(0,0) = f_2^k(a, b)\).
(b.) For any real number \(a\) and any positive integer, there is at most one solution \(y\) for \(f_1^k(0,0) = f_2^k(a, y)\).
(c.) There exists \((a, b) \in \mathbb{R}^2\) such that \(f_1^k(a, b) \neq f_2^k(x, y)\) for any \((x, y) \in \mathbb{R}^2\) and any positive integer \(k\).
(d.) \(f_1\) is a linear transformation.

6. Let \(f: \mathbb{R}^n \rightarrow \mathbb{R}\) be the map \(f\left(x_1, \ldots, x_n\right)=a_1 x_1+\ldots+a_n x_n\), where \(a=\left(a_1, \ldots, a_n\right)\) is a fixed non-zero vector. Let \(D f(0)\) denote the derivative of \(f\) at 0. Which of the following are correct?

(a.) \((D f)(0)\) is a linear map from \(\mathbb{R}^*\) to \(\mathbb{R}\)
(b.) \([(D f)(0)](a)=|a|^2\)
(c.) \([(D f)(0)](a)=0\)
(d.) \([(D f)(0)](b)=a_1 b_1+\ldots+a_n b_n\) for \(b=\left(b_1, \ldots, b_n\right)\)

7. Let \(f(x, y)=\sqrt{|xy|}\). Then,

(a.) \(f_x\) and \(f_y\) do not exist at \((0,0)\)
(b.) \(f_x(0,0)=1\)
(c.) \(f_y(0,0)=0\)
(d.) \(f\) is differentiable at \((0,0)\)

8. Let \(z=x+iy\) and \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be the function \(f(x, y)=z^2=\left(x^2-y^2, 2xy\right) \in \mathbb{R}^2\). Let \((D f)(a)\) denote the derivative of \(f\) at \(a\). Which of the following are true?

(a.) \((D f)(a)h=2ah\), where \(a=a_1+ia_2\) and \(h=h_1+ih_2\)
(b.) \((D f)(a)=2\left(\begin{array}{cc}a_1 & -a_2 \\ a_2 & a_1\end{array}\right)\), \(a=\left(a_1, a_2\right) \in \mathbb{R}^2\)
(c.) \(f\) is one-to-one on \(\mathbb{R}^2\)
(d.) For any \(a \in \mathbb{R}^2 /\{(0,0)\}\), \(f\) is one-to-one on some neighborhood of \(a\)

9. Let \(f: \mathbb{R}^n \rightarrow \mathbb{R}^n\) be the function defined by \(f(x)=x \|\left. x\right|^2\) for \(x \in \mathbb{R}^n\). Which of the following statements are correct?

(a.) \((D f)(0)=0\)
(b.) \((D f)(x)=0\) for all \(x \in \mathbf{R}^*\)
(c.) \(f\) is one-to-one
(d.) \(f\) has an inverse

10. Let \(f: A \cup E \rightarrow \mathbb{R}^2\) be differentiable, where \(A=\left\{(x, y) \in \mathbb{R}^2: \frac{1}{2}

(a.) If \((D f)(x, y)=0\) for all \((x, y) \in A \cup E\), then \(f\) is constant
(b.) If \((D f)(x, y)=0\) for all \((x, y) \in A\), then \(f\) is constant on \(A\)
(c.) If \((D f)(x, y)=0\) for all \((x, y) \in E\), then \(f\) is constant on \(E\)
(d.) If \((D f)(x, y)=0\) for all \((x, y) \in A \cup E\), then for some \((x_0, y_0),(x_1, y_1) \in \mathbf{R}^2\), \(f(x, y)=(x_0, y_0)\) for all \((x, y) \in A\) and \(f(x, y)=(x_1, y_1)\) for all \((x, y) \in E\)

11. Let \(L: \mathbf{R}^* \rightarrow \mathbf{R}\) be the function \(L(x)=\langle x, y\rangle\), where \(\langle\cdot, \cdot\rangle\) is some inner product on \(\mathbb{R}^*\) and \(y\) is a fixed vector in \(\mathbb{R}^*\). Further denote by \(D L\), the derivative of \(L\). Which of the following are necessarily correct?

(a.) \(D L(u)=D L(v)\) for all \(u, v \in \mathbb{R}^*\)
(b.) \(D L(0,0, \ldots, 0)=L\)
(c.) \(D L(x)=\|x\|^2\) for all \(x \in \mathbb{R}^*\)
(d.) \(D L(1,1, \ldots, 1)=0\)

12. Let \(f:[\pi, 2 \pi] \rightarrow \mathbb{R}^2\) be the function \(f(t)=(\cos t, \sin t)\). Which of the following are necessarily correct?

(a.) There exists \(t_0 \in[\pi, 2 \pi]\) such that \(f^{\prime}\left(t_0\right)=\frac{1}{\pi}(f(2 \pi)-f(\pi))\)
(b.) There does not exist any \(t_0 \in[\pi, 2 \pi]\) such that \(f^{\prime}\left(t_0\right)=\frac{1}{\pi}(f(2 \pi)-f(\pi))\)
(c.) There exists \(t_0 \in[\pi, 2 \pi]\) such that \(|f(2 \pi)-f(\pi)| \leq \pi| f^{\prime}\left(t_0\right) |\)
(d.) \(f^{\prime}(t)=(-\sin t, \cos t)\) for all \(t \in[\pi, 2 \pi]\)

13. Consider the map \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) defined by \(f(x, y)=\left(7 x+x^4, 3 x+4 y+y^4\right)\). Then,

(a.) \(f\) is discontinuous at \((0,0)\)
(b.) \(f\) is continuous at \((0,0)\) and directional derivatives exist at \((0,0)\)
(c.) \(f\) is differentiable at \((0,0)\) but the derivative \(D f(0,0)\) is not invertible
(d.) \(f\) is differentiable at \((0,0)\) and the derivative \(D f(0,0)\) is invertible

14. The map \(L: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) given \(L(x, y)=(x,-y)\) is

(a.) Differentiable everywhere on \(\mathbb{R}^2\)
(b.) Differentiable only at \((0,0)\)
(c.) \(D L(0,0)=(0,0)\)
(d.) \(D L(0,0)=(1,-1)\

15. Consider the map \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) defined by \(f(x, y)=\left(3 x-2 y+x^2, 4 x+5 y+y^2\right)\). Then

(a.) \(f\) is discontinuous at \((0,0)\)
(b.) \(f\) is continuous at \((0,0)\) and all directional derivatives exist at \((0,0)\)
(c.) \(f\) is differentiable at \((0,0)\) but the derivative \(D f(0,0)\) is not invertible
(d.) \(f\) is differentiable at \((0,0)\) and the derivative \(D f(0,0)\) is invertible

16. Let \(f: \mathbb{R}^2 \rightarrow \mathbb{R}\) and \(g: \mathbb{R}^2 \rightarrow \mathbb{R}\) be defined by \(f(x, y)=|x|+|y|\) and \(g(x, y)=|x y|\). Then

(a.) \(f\) is differentiable at \((0,0)\), but \(g\) is not differentiable at \((0,0)\)
(b.) \(g\) is differentiable at \((0,0)\), but \(f\) is not differentiable at \((0,0)\)
(c.) Both \(f\) and \(g\) are differentiable at \((0,0)\)
(d.) Both \(f\) and \(g\) are continuous at \((0,0)\)

17. Let

\(f(x, y)= \begin{cases}\frac{x y}{\left(x^2+y^2\right)^{3 / 2}}\left[1-\cos \left(x^2+y^2\right)\right], & (x, y) \neq(0,0) \\ k, & (x, y)=(0,0)\end{cases}\)

Then the value of \(k\) for which \(f(x, y)\) is continuous at \((0,0)\) is

(a.) 0
(b.) \(\frac{1}{2}\)
(c.) 1
(d.) \(\frac{3}{2}\)

18. For \((x, y) \in \mathbb{R}^2\), \(f(x, y)= \begin{cases}\frac{2 x y}{x^2+y^2}, & \text { if }(x, y) \neq(0,0) \\ 0, & \text { if }(x, y)=(0,0)\end{cases}\) Then

(a.) \(f_x\) and \(f_y\) exist at \((0,0)\), \(f\) is continuous at \((0,0)\)
(b.) \(f_x\) and \(f_y\) exist at \((0,0)\) and \(f\) is discontinuous at \((0,0)\)
(c.) \(f_x\) and \(f_y\) do not exist at \((0,0)\), \(f\) is continuous at \((0,0)\)
(d.) \(f_x\) and \(f_y\) do not exist at \((0,0)\), \(f\) is discontinuous at \((0,0)\)

19. Let \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be defined by \(f(x, y)=\left\{\begin{array}{cc}x^2+y^2 & \text { If } \mathrm{x} \text { and } \mathrm{y} \text { are rational } \\ 0 & \text { Otherwise }\end{array}\right.\) Then

(a.) \(f\) is not continuous at \((0,0)\)
(b.) \(f\) is continuous at \((0,0)\) but not differentiable at \((0,0)\)
(c.) \(f\) is differentiable only at \((0,0)\)
(d.) \(f\) is differentiable everywhere

20. If \(f(x, y)=\left\{\begin{array}{cc}\frac{x^3}{x^2+y^2}, & \text { if }(x, y) \neq(0,0) \\ 0, & \text { if }(x, y)=(0,0)\end{array}\right.\), then at \((0,0)\)

(a.) \(f_x\), \(f_y\) do not exist
(b.) \(f_x\), \(f_y\) exist and are equal
(c.) The directional derivative exists along any straight line
(d.) \(f\) is differentiable

21. Let

\(f(x, y)=\frac{x y}{\left(x^2+y^2\right)^{3 / 2}}\left[1-\cos \left(x^2+y^2\right)\right], \quad (x, y) \neq(0,0)\)

\(f(x, y)=k, \quad (x, y)=(0,0)\)

Then the value of \(k\) for which \(f(x, y)\) is continuous at \((0,0)\) is

(a.) 0
(b.) \(\frac{1}{2}\)
(c.) 1
(d.) \(\frac{3}{2}\)

22. Consider the function \(f(x, y)=\frac{x^2}{y^2}, \quad (x, y) \in\left[\frac{1}{2}, \frac{3}{2}\right] \times\left[\frac{1}{2}, \frac{3}{2}\right]\). The derivative of the function at \((1,1)\) along the direction \((1,1)\)

(a.) 0
(b.) 1
(c.) 2
(d.) -2

23. Let \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be the function \(f(r, \theta)=(r \cos \theta, r \sin \theta)\). Then for which of the open subsets \(U\) of \(\mathbb{R}^2\) given below, \(f\) restricted to \(U\) admits an inverse?

(a.) \(U=\mathbb{R}^2\)
(b.) \(U=\left\{(x, y) \in \mathbb{R}^2: x>0, y>0\right\}\)
(c.) \(U=\left\{(x, y) \in \mathbb{R}^2: x^2+y^2<1\right\}\)
(d.) \(U=\left\{(x, y) \in \mathbb{R}^2: x<-1, y<-1\right\}\)

24. Let \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be given by \(f(x, y)=\left(x^2, y^2+\sin x\right)\). Then the derivative of \(f\) at \((x, y)\) is the linear transformation given by

(a.) \(\left(\begin{array}{cc}2 x & 0 \\ \cos x & 2 y\end{array}\right)\)
(b.) \(\left(\begin{array}{cc}2 x & 0 \\ 2 y & \cos x\end{array}\right)\)
(c.) \(\left(\begin{array}{cc}2 y & \cos x \\ 2 x & 0\end{array}\right)\)
(d.) \(\left(\begin{array}{cc}2 x & 2 y \\ 0 & \cos x\end{array}\right)\)

25. A function \(f: \mathbb{R}^2 \rightarrow \mathbb{R}\) is defined by \(f(x, y)=x y\). Let \(v=(1,2)\) and \(a=\left(a_1, a_2\right)\) be two elements of \(\mathbb{R}^2\). The directional derivative of \(f\) in the direction of \(v\) at \(a\) is

(a.) \(a_1+2 a_2\)
(b.) \(a_2+2 a_1\)
(c.) \(\frac{a_2}{2}+a_1\)
(d.) \(\frac{a_1}{2}+a_2\)

26. A function \(f(x, y)\) on \(\mathbb{R}^2\) has the following partial derivatives

\(\frac{\partial f}{\partial x}(x, y)=x^2, \quad \frac{\partial f}{\partial y}(x, y)=y^2\)

Then

(a.) \(f\) has directional derivatives in all direction everywhere.
(b.) \(f\) has a derivative at all points.
(c.) \(f\) has directional derivative only along the direction \((1,1)\) everywhere.
(d.) \(f\) does not have directional derivatives in any direction everywhere.

27. Suppose that \(f: \mathbb{R}^* \rightarrow \mathbb{R}\) is given by \(f(\underline{x})=a_1 x_1^2+a_2 x_2^2+\cdots+a_n x_n^2\), where \(\underline{x}=\left(x_1, x_2, \ldots, x_n\right)\) and at least one \(a_j\) is not zero. Then we can conclude that

(a.) \(f\) is not everywhere differentiable
(b.) the gradient \((\nabla f)(\underline{x}) \neq 0\) for every \(\underline{x} \in \mathbb{R}^n\)
(c.) if \(\underline{x} \in \mathbb{R}^n\) is such that \((\nabla f)(\underline{x})=0\) then \(f(\underline{x})=0\)
(d.) if \(\underline{x} \in \mathbb{R}^n\) is such that \(f(\underline{x})=0\) then \((\nabla f)(\underline{x})=0\)

28. Let \(S\) be the set of \((\alpha, \beta) \in \mathbb{R}^2\) such that \(\frac{x^a y^b}{\sqrt{x^2+y^2}} \rightarrow 0\) as \((x, y) \rightarrow(0,0)\)

Then \(S\) is contained in

(a.) \(\{(\alpha, \beta): \alpha>0, \beta>0\}\)
(b.) \(\{(\alpha, \beta): \alpha>2, \beta>2\}\)
(c.) \(\{(\alpha, \beta): \alpha+\beta>1\}\)
(d.) \(\{(\alpha, \beta): \alpha+4 \beta>1\}\)

29. Let \(f(x, y)=\frac{1-\cos (x+y)}{x^2+y^2}\) if \((x, y) \neq(0,0)\), \(f(0,0)=\frac{1}{2}\) and \(g(x, y)=\frac{1-\cos (x+y)}{(x+y)}\) if \(x+y \neq 0\), \(g(x, y)=\frac{1}{2}\) if \(x+y=0\) then

(a.) \(f\) is continuous at \((0,0)\)
(b.) \(f\) is continuous everywhere except at \((0,0)\)
(c.) \(g\) is continuous at \((0,0)\)
(d.) \(g\) is continuous everywhere

30. Let \(f: \mathbb{R}^4 \rightarrow \mathbb{R}\) be a function defined by \(f(x)=x^t A x\), where \(A\) is a \(4 \times 4\) matrix with real entries and \(x^t\) denotes the transpose of \(x\). The gradient of \(f\) at a point \(x_0\) necessarily is

(a.) \(2 A x_0\)
(b.) \(A x_0+A^t x_0\)
(c.) \(2 A^t x_0\)
(d.) \(A x_0\)

31. Let \(f: \mathbb{R}^n \rightarrow \mathbb{R}^n\) be a continuously differentiable map satisfying \(\|f(x)-f(y)\| \geq\|x-y\|\), for \(x, y \in \mathbb{R}^n\). Then

(a.) \(f\) is onto
(b.) \(f(\mathbb{R}^n)\) is a closed subset of \(\mathbb{R}^n\)
(c.) \(f(\mathbb{R}^n)\) is an open subset of \(\mathbb{R}^n\)
(d.) \(f(0)=0\)

32. Let \(A\) be a connected open subset of \(\mathbb{R}^2\). The number of continuous surjective functions from \(\bar{A}\) (the closure of \(A\) in \(\mathbb{R}^2\) ) to \(\mathbb{Q}\) is:

(a.) 1
(b.) 0
(c.) 2
(d.) Not finite