Question

Let f, g and h be real valued functions defined on the interval [0, 1] by f(x) = e^x2 + e^{– x2} , g(x) = xe^x2 + e^{– x2} and h(x) = x² e^x2 + e^{– x2} . If a, b and c denote respectively the absolute maximum on [0, 1] then

(a) a = b, c ≠ b

(b) a = c, a ≠ b

(c) a ≠ b, c ≠ a

(d) a = b = c

Answer 1

Correct option (d) a = b = c

Explanation:

Given f(x) = e^x2 + e^–x2

⇒ f'(x) = 2x(e^x2 – e^–x2 ) ≤ 0, ∀ x ∈ [0, 1]

Also, g'(x) = e^x2 + 2x² e^x2 – 2xe^–x2 ≥ 0

for all x in [0, 1]

and h'(x) = 2xe^x2 + 2x³e^x2 – 2xe^–x2 ≥ 0

for all x in [0, 1]

Clearly, for 0 ≤ x ≤ 1, f(x) ≥ g(x) ≥ h(x)

Thus, f(1) = g(1) = h(1) = e + 1/e

⇒ a = b = c