Consider the function \(f(x)=x^3-3x\). Then
Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | Point of local Maxima | (I) | 1 |
| (B) | Point of local Minima | (II) | -1 |
| (C) | Local maximum value | (III) | 2 |
| (D) | Local minimum value | (IV) | -2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(II), (B)-(I), (C)-(III), (D)-(IV)
The critical points come from \(f'(x) = 0\):
\(f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1)\)
\(f'(x) = 0 \Rightarrow x = -1 \text{ or } x = 1\)
\(f''(x) = 6x\)
By the second derivative test:
(A) Point of local Maxima
This matches (II) −1, since \(f''(-1) = -6 \lt 0\).
(B) Point of local Minima
This matches (I) 1, since \(f''(1) = 6 \gt 0\).
(C) Local maximum value
This matches (III) 2:
\(f(-1) = -1 + 3 = 2\)
(D) Local minimum value
This matches (IV) −2:
\(f(1) = 1 - 3 = -2\)
So the matching is (A)-(II), (B)-(I), (C)-(III), (D)-(IV).
Tip: The point of local maxima is the x-value (here −1); the local maximum value is f at that point (here 2). Option (3) swaps the points and the values.
NCERT (2026-27 edition):
Maxima and Minima
