Functions of several real variables - Calculus - Statistics Previous Year Questions

1

2019 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2019

The maximum value of \((x - 1)^2 + (y - 2)^2\) subject to the constraint \(x^2 + y^2 \leq 45\) is equal to...

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2

2020 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2020

Let \(S = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x^2 - y^2 = 4\}\) and \(f : S \to \mathbb{R}\) be defined by \(f(x,y) = 6x + y^2\), where \(\mathbb{R}\) denotes the set of all real numbers. Then

A

\(f\) is bounded on \(S\)

B

the maximum value of \(f\) on \(S\) is 13

C

the minimum value of \(f\) on \(S\) is \(-14\)

D

the minimum value of \(f\) on \(S\) is \(-13\)

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3

2020 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2020

Let \(f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) be defined by \(f(x,y) = \begin{cases} \frac{4x^2 + 9y^2}{\sqrt{4x^2 + 9y^2 + 64} - 8}, & (x,y) \neq (0,0) \\ c, & (x,y) = (0,0) \end{cases}\), where \(\mathbb{R}\) denotes the set of all real numbers and \(c \in \mathbb{R}\) is a fixed constant. Then, which of the following statements is TRUE?

A

There does NOT exist a value of \(c\) for which \(f\) is continuous at \((0,0)\)

B

\(f\) is continuous at \((0,0)\) if \(c = 0\)

C

\(f\) is continuous at \((0,0)\) if \(c = 10\)

D

\(f\) is continuous at \((0,0)\) if \(c = 16\)

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4

2020 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2020

Let \( f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) be defined by
\[ f(x, y) = x^4 - 2x^2y + 16y + 17, \]
where \( \mathbb{R} \) denotes the set of all real numbers. Then

A

\(f \) has a local minimum at \((2, \frac{3}{2}) \)

B

\(f \) has a local maximum at \((2, \frac{3}{2}) \)

C

\(f \) has a saddle point at \((2, \frac{3}{2}) \)

D

\(f \) is bounded

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5

2020 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2020

Let \( f : \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) be defined by \( f(x, y) = |y - 2| \sqrt{|x - 1|} \), \( (x, y) \in \mathbb{R} \times \mathbb{R} \), where \( \mathbb{R} \) denotes the set of all real numbers. Then which of the following statements is TRUE?

A

\(f \) is differentiable at \((1, 2) \)

B

\(f \) is continuous at \((1, 2) \) but NOT differentiable at \((1, 2) \)

C

The partial derivative of \(f \) with respect to \(x \), at \((1, 2) \) does NOT exist

D

The directional derivative of \(f \) at \((1, 2) \) along \(\vec{u} = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}) \) equals 1

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6

2021 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2021

Let \(f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) be defined by \(f(x, y) = 8x^2 - 2y\), where \(\mathbb{R}\) denotes the set of all real numbers. If \(M\) and \(m\) denote the maximum and minimum values of \(f\), respectively, on the set \(\{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 1\}\), then \(M - m\) equals __________ (round off to 2 decimal places).

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7

2022 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2022

Let \(f: \mathbb{R}^2 \to \mathbb{R}\) be a function defined by
\[f(x, y) = \begin{cases} \frac{x^2 y}{x^2 + y^2}, & (x, y) \neq (0,0), \\ 0, & (x, y) = (0,0). \end{cases}\]
Then which one of the following statements is true?

A

\(f\) is bounded and \(\frac{\partial f}{\partial x}\) is unbounded on \(\mathbb{R}^2\)

B

\(f\) is unbounded and \(\frac{\partial f}{\partial x}\) is bounded on \(\mathbb{R}^2\)

C

Both \(f\) and \(\frac{\partial f}{\partial x}\) are unbounded on \(\mathbb{R}^2\)

D

Both \(f\) and \(\frac{\partial f}{\partial x}\) are bounded on \(\mathbb{R}^2\)

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8

2022 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2022

If \(P(x, y, z)\) is a point which is nearest to the origin and lies on the intersection of the surfaces \(z = xy + 5\) and \(x + y + z = 1\). Then the distance (in integer) between the origin and the point \(P\) is ________

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9

2023 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2023

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be defined by \( f(x, y) = xy \). Then the maximum value (rounded off to two decimal places) of \( f \) on the ellipse \( x^2 + 2y^2 = 1 \) equals ______________

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10

2024 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2024

Consider the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by
\( f(x, y) = 108xy - 2x^2y - 2xy^2. \)
Which one of the following statements is NOT true?

A

\(f \) has four critical points

B

\(f \) has a local minimum at \((0, 0) \)

C

\(f \) has a local maximum at \((18, 18) \)

D

\(f \) has two or more saddle points

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11

2024 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2024

Consider the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by
\[ f(x,y) = \begin{cases} \frac{x^3 - y^3}{x^2 + y^2} & \text{if } (x,y) \neq (0,0) \\ 0 & \text{otherwise.} \end{cases} \]
If \( f_x \) denotes the partial derivative of \( f \) with respect to \( x \) and \( f_y \) denotes the partial derivative of \( f \) with respect to \( y \), then which one of the following statements is NOT true?

A

\(f \) is continuous at \((0,0) \)

B

\(f_x(0,0) \neq f_y(0,0) \)

C

\(f_x \) is continuous at \((0,0) \)

D

\(f_y \) is not continuous at \((0,0) \)

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12

2025 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2025

Let \(U = \{(x, y) \in \mathbb{R}^2: x + y \le 2\}\). Define \(f: U \to \mathbb{R}\) by \[ f(x, y) = (x - 1)^4 + (y - 2)^4. \] The minimum value of \(f\) over \(U\) is

A

0

B

\(\frac{1}{16}\)

C

\(\frac{17}{81}\)

D

\(\frac{1}{8}\)

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13

2025 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2025

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be defined as $f(x,y) = x^2 y^2 + 8x - 4y$. The number of saddle points of $f$ is ______. (answer in integer).

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14

2026 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2026

Consider the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined by \[ f(x_1, x_2) = 2x_1^4 + x_2^2 + x_2x_1^2. \] Which of the following statements is correct?

A

There are two distinct stationary points of \(f \)

B

\((0, 0) \) is a saddle point of \(f \)

C

\((0, 0) \) is a local maximum of \(f \)

D

\((0, 0) \) is a local minimum of \(f \)

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15

2026 · Statistics · Calculus · Functions of several real variables

Statistics (ST) 2026

Consider the function \(f: \mathbb{R}^2 \to \mathbb{R}\) defined by
\[ f(x, y) = \begin{cases} \dfrac{x^3 + y^3}{\sqrt{x^2 + 2y^2}} & (x, y) \neq 0 \\ 0 & (x, y) = 0. \end{cases} \]
Which of the following statements is/are correct?

A

\(f\) is continuous at \((0, 0)\)

B

Partial derivatives \(f_x\) and \(f_y\) exist at \((0, 0)\) and \(f_x(0, 0) = 0\), \(f_y(0, 0) = 0\)

C

\(f\) is not differentiable at \((0, 0)\)

D

\(f\) is differentiable

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