Question

If f(x) = ex, 0 ≤ x ≤ 1

f(x) = 2 - ex-1, 1 < x ≤ 2

f(x) = x - e, 2 < x ≤ 3

and g(x) = ∫ f(t) (for x → 0,x)dt, x ∈ [1, 3], then

(a) g(x) has local maxima at x = 1 + loge2 and local minima at x = e

(b) f(x) has local maxima at x = 1 and local minima at x = 2

(c) g(x) has no local minima

(d) f(x) has no local maxima

Answer 1

Correct option

(a) g(x) has local maxima at x = 1 + loge2 and local minima at x = e

(b) f(x) has local maxima at x = 1 and local minima at x = 2

Explanation :

has a local maximum.

Also, at x = e,

g"(e) = 1 > 0, g(x) has a local minima.

'.' f(x ) is discontinuous at x = 1, then we get local maxima x = 1 and local minima at x = 2.

Hence, (a) and (b) are correct answers.