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CGP EDU Academic Team
Published on: August 14, 2026
Let f and g be two differentiable functions defined on an interval Ι such that f(x) ≥ 0 and g(x) ≤ 0 for all x ∈ Ι and f is strictly decreasing on Ι while g is strictly increasing on Ι then
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
(a,d)Let h(x) = f(x) g(x)
h ′ (x) = f ′ (x) g(x) + g ′ (x) f(x)
As f ′ (x) < 0, g(x) ≤ 0 ⇒ f ′ (x) g(x) ≥ 0
and g ′ (x) > 0, f(x) ≥ 0 ⇒ f(x) g ′ (x) ≥ 0
⇒ h ′ (x) ≥ 0
⇒ h(x) is increasing.
Let x 1 , x 2 ∈ Ι
x 1 < x 2
g(x 1 ) < g(x 2 )
f(g(x 1 )) > f(g(x 2 ))
fog(x 1 ) > fog(x 2 )
⇒ fog(x) is monotonically decreasing.
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