1
Consider the function f : (0, ) (- , ) given by f(x) = x e(x) - x + 1 . Then which one of the following statements is TRUE ?
- The derivative of the function f is decreasing in the interval (0, 1)
- The function f has a local maximum at some point a (0, )
- The function f has a local minimum at some point b (0, )
- The function f has NEITHER a point of local maximum NOR a point of local minimum in the interval (0, )
Answer
(4) The function f has NEITHER a point of local maximum NOR a point of local minimum in the interval (0, )
Solution
Find the sign of f'(x) on the whole domain.
f'(x) = x2 x + 1 x - 1 = x + 2 - 2 x2 x
Let g(x) = x + 2 - 2 x; g'(x) = 1x - 1 x = 1 - xx, so g increases on (0, 1) and decreases on (1, ).
Maximum of g is g(1) = 0, so g(x) 0, i.e. f'(x) 0 for all x > 0 (zero only at x = 1).
Hence f is strictly decreasing: no local maximum and no local minimum.
Also f''(x) = - x4x x > 0 on (0, 1), so f' is increasing there; (A) is false.
Answer: (D)
JEE Advanced 2026 Paper 1