Let f be continuous on [a, b] and twice differentiable on (a, b), and let c be a point of (a, b). Match List-I with List-II
| List-I (Condition) | List-II (Conclusion) | ||
| (A) | \(f'(c) = 0\) and \(f''(c) \lt 0\) | (I) | f is increasing in [a, b] |
| (B) | \(f'(c) = 0\) and \(f''(c) \gt 0\) | (II) | x = c is a point of local minima |
| (C) | \(f'(c) = 0\) and \(f''(c) = 0\) | (III) | The second derivative test fails |
| (D) | \(f'(x) \gt 0\) for each x in (a, b) | (IV) | x = c is a point of local maxima |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
These are the second derivative test and the basic result on increasing functions.
(A) \(f'(c) = 0\) and \(f''(c) \lt 0\)
This matches (IV) x = c is a point of local maxima.
f′(c) = 0 and f″(c) < 0: f′ is decreasing through c, so it goes from + to −; for example \(f(x) = -x^{2}\) at c = 0
(B) \(f'(c) = 0\) and \(f''(c) \gt 0\)
This matches (II) x = c is a point of local minima.
f′(c) = 0 and f″(c) > 0: f′ goes from − to + through c; for example \(f(x) = x^{2}\) at c = 0
(C) \(f'(c) = 0\) and \(f''(c) = 0\)
This matches (III) The second derivative test fails.
Nothing can be concluded: at c = 0, \(x^{4}\) has a minimum, \(-x^{4}\) a maximum and \(x^{3}\) neither
(D) \(f'(x) \gt 0\) for each x in (a, b)
This matches (I) f is increasing in [a, b].
A positive derivative throughout (a, b), with f continuous on [a, b], makes f increasing in [a, b]