Question

If f(x) is a function which is derivable in an interval 1 containing a point c, then match List I with List II.

List I		List II
A.	f(x) has second order derivate at x = c such that f'(c) = 0 and f'(c) < 0; then	I.	point of inflexion of f(x)
B.	Necessary condition for point x = c to be extreme point of f(x) is	II.	‘c’ is point of local minima of f(x)
C.	If f'(x) does not change its sign as x crosses the point x = c then it is called a	III.	c is a critical point of f(x)
D.	f(x) has second order derivate at x = c such that f'(c) and f'(c) > 0; then	IV.	‘c’ is point of local maxima of f(x)

Choose the correct answer from the options given below :

• A. A-IV, B-I, C-III, D-II
• B. A-II, B-I, C-III, D-IV
• C. A-II, B-III, C-I, D-IV
• D. A-IV, B-III, C-I, D-II

Answer 1

The Correct Option is D

In this problem, we need to match statements from List I with their corresponding statements in List II regarding characteristics of function f(x) at a point c.

Let's analyze each pair:

• A: f(x) has second order derivative at x = c such that f'(c) = 0 and f'(c) < 0; then
  This statement is examining the concavity of the function at point c. Since f'(c) = 0 and f''(c) < 0, x = c is a point where the function has a local maximum. Thus, the correct match is IV: ‘c’ is point of local maxima of f(x).
• B: Necessary condition for point x = c to be extreme point of f(x) is
  For a point to be considered an extreme point (local minima or maxima), the first derivative f'(c) should be zero or undefined. This makes c a critical point. Therefore, the correct match is III: c is a critical point of f(x).
• C: If f'(x) does not change its sign as x crosses the point x = c then it is called a
  This describes the behavior of the derivative when c is a point of inflection. Here, the function neither reaches a peak nor a trough but changes curvature. Thus, the correct match is I: point of inflexion of f(x).
• D: f(x) has second order derivative at x = c such that f''(c) and f'(c) > 0; then
  Given f'(c) > 0 and f''(c) indicates the concavity, but since f'(c) > 0 by itself doesn't imply behavior related to concavity for extreme determination, we have to pair with its effect which is concave upwards, commonly indicating a local minimum. Therefore, the correct match is II: ‘c’ is point of local minima of f(x).

Thus, the correct answer is A-IV, B-III, C-I, D-II.