SYLLABUS
Functions of a single variable, limit, continuity and differentiability, Taylor series, maxima and minima, optimization involving a single variable.
Q1 – For any twice differentiable function 𝑓:ℝ→ℝ, if at some 𝑥* ∈ ℝ, 𝑓′(𝑥*) = 0 and 𝑓′′(𝑥*) > 0, then the function 𝑓 necessarily has a ______ at 𝑥 = 𝑥*.
Note: ℝ denotes the set of real numbers.
(A) local minimum
(B) global minimum
(C) local maximum
(D) global maximum
(GATE DS&AI 2024)
Ans – (A)
Explanation – For a function f, if we have an f that is twice differentiable, using the first and second derivative tests can give us insight on how this function behaves. If at some point x*, the first derivative f′(x) = 0, this implies that the function has a flat slope there, and is neither increasing nor decreasing at the exact point. Now, if the second derivative f′′(x*) > 0, we have a bowl that opens towards y axis. This is evidence by the curve rising as it goes up, meaning the point x* is a Local Minimum, a point at which the function value is lower than the values of all neighbouring points. So, the answer will be option (A) a local minimum.
Q2 – Consider the function 𝑓:ℝ→ℝ where ℝ is the set of all real numbers. 𝑓(𝑥) = 𝑥4/4 − 2𝑥3/3 − 3𝑥2/2 + 1
Which of the following statements is/are TRUE?
(A) 𝑥=0 is a local maximum of f
(B) 𝑥=3 is a local minimum of f
(C) 𝑥=−1 is a local maximum of f
(D) 𝑥=0 is a local minimum of f
(GATE DS&AI 2024)
Ans – (A, B)
Explanation – Polynomial f(x) = 𝑥4/4 − 2𝑥3/3 − 3𝑥2/2 + 1
First Derivative f’(x) = x3 – 2x2 – 3x
- x(x2 – 2x – 3)
- x = 0, x = 3, x = -1
Second Derivative f’’(x) = 3x2 – 4x – 3
- f’’(0) = -3 and -3 < 0 local Maximum of f.
- f’’(3) = 12 and 12 > 0 local Minimum of f.
- f’’(-1) = 4 and 4 > 0 local Minimum of f.
So, option A and B is correct.
Q3 – Evaluate the following limit:
(GATE DS&AI 2024)
Ans – (0.5)
Explanation – 