Calculus and Optimization - Data Science & Artificial Intelligence Previous Year Questions

1

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

For any twice differentiable function \( f: \mathbb{R} \to \mathbb{R} \), if at some \( x^* \in \mathbb{R} \), \( f'(x^*) = 0 \) and \( f''(x^*) > 0 \), then the function \( f \) necessarily has a ______ at \( x = x^* \).
Note: \( \mathbb{R} \) denotes the set of real numbers.

A

local minimum

B

global minimum

C

local maximum

D

global maximum

Open complete paper

2

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

Let \(f(x) = \frac{e^x - e^{-x}}{2}\), \(x \in \mathbb{R}\). Let \(f^{(k)}(a)\) denote the \(k^{th}\) derivative of \(f\) evaluated at \(a\). What is the value of \(f^{(10)}(0)\)? (Note: ! denotes factorial)

A

0

B

1

C

\(\frac{1}{10!}\)

D

\(\frac{2}{10!}\)

Open complete paper

3

2026 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science and Artificial Intelligence (DA) 2026

Let \( f(x) = x^3 - 3x^2 + 2 \) be a function defined on \( (-1, 3] \).
Which of the following statements is/are correct?

A

\(f(x) \) has exactly two roots in \([-0.9, 0] \).

B

\(f(x) \) has a minimum at 2 only.

C

\(f(x) \) has maximum at 0 only.

D

\(f(x) \) has a root at 1.

Check answer

Open complete paper

4

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

Let f: ℝ → ℝ be the function f(x) = 1 / (1 + e^{-x}).
The value of the derivative of f at x where f(x) = 0.4 is ______ (rounded off to two decimal places).
Note: ℝ denotes the set of real numbers.

Check answer

Open complete paper

5

2025 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2025

Consider two functions \(f : \mathbb{R} \to \mathbb{R}\) and \(g : \mathbb{R} \to (1, \infty)\). Both functions are differentiable at a point \(c\). Which of the following functions is/are ALWAYS differentiable at \(c\)? The symbol \(\cdot\) denotes product and the symbol \(\circ\) denotes composition of functions.

A

\(f \pm g\)

B

\(f \cdot g\)

C

\(\frac{f}{g}\)

D

\(f \circ g + g \circ f\)

Check answer

Open complete paper

6

2024 · Data Science & Artificial Intelligence · Calculus and Optimization · Single-variable Differential Calculus

Data Science & Artificial Intelligence (DA) 2024

Let \(f : \mathbb{R} \to \mathbb{R}\) be a function. Note: \(\mathbb{R}\) denotes the set of real numbers.
\(f(x) = \begin{cases} -x, & \text{if } x < -2 \\ ax^2 + bx + c, & \text{if } x \in [-2, 2] \\ x, & \text{if } x > 2 \end{cases}\)
Which ONE of the following choices gives the values of \(a, b, c\) that make the function \(f\) continuous and differentiable?

A

\(a = \frac{1}{4}, b = 0, c = 1\)

B

\(a = \frac{1}{2}, b = 0, c = 0\)

C

\(a = 0, b = 0, c = 0\)

D

\(a = 1, b = 1, c = -4\)

Open complete paper