If \(f(x)\) is a function which is derivable in an interval I containing a point c, then match List I with List II.
| LIST I | LIST II | ||
| A. | \(f(x)\) has second order derivative at \(x = c\) such that \(f'(c) = 0\) and \(f''(c) \lt 0\); 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 derivative at \(x = c\) such that \(f'(c)\) and \(f''(c) \gt 0\); then | IV. | ‘c’ is point of local maxima of \(f(x)\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A-IV, B-III, C-I, D-II
(A) \(f'(c) = 0\) and \(f''(c) \lt 0\)
This matches (IV) ‘c’ is point of local maxima of \(f(x)\).
By the second derivative test, \(f''(c) \lt 0\) at a critical point means the graph bends downwards there, so c is a local maximum.
(B) Necessary condition for an extreme point
This matches (III) c is a critical point of \(f(x)\).
For a derivable function, a local maximum or minimum at c forces \(f'(c) = 0\), that is, c must be a critical point. (The condition is necessary, not sufficient.)
(C) \(f'(x)\) does not change sign as x crosses c
This matches (I) point of inflexion of \(f(x)\).
By the first derivative test, if \(f'\) keeps the same sign on both sides, f has neither a maximum nor a minimum at c; such a point is called a point of inflexion.
(D) \(f'(c) = 0\) and \(f''(c) \gt 0\)
This matches (II) ‘c’ is point of local minima of \(f(x)\).
\(f''(c) \gt 0\) at a critical point means the graph bends upwards there, so c is a local minimum.
Note: It seems that there is a mistake in the original NTA paper, which prints item D as “\(f'(c)\) and \(f''(c) \gt 0\)” when it should read “\(f'(c) = 0\) and \(f''(c) \gt 0\)”. The question can still be solved, because items A, B and C have only one possible match each, and Option (4) is the only option containing A-IV, B-III and C-I.