Question

The point in the interval [0, 2π], where f(x) = e^x sin x has maximum slope, is ______.

पर्याय

• `π/4`

• `π/2`

• π

• `(3π)/2`

Answer 1

The point in the interval [0, 2π], where f(x) = e^x sin x has maximum slope, is `underlinebb(π/2)`.

Explanation:

Given, f(x) = e^x sin x

f'(x) = e^x cos x + sin x e^x

f''(x) = – e^x sin x + cos x e^x + e^x cos x + e^x sin x

= 2e^x cos x

For maximum slope,

f''(x) = 0

`\implies` 2e^x cos x = 0

`\implies` cos x = 0

`\implies` x = `π/2, (3π)/2, ∀  x ∈ [0, 2π]`

f'''(x) = 2 [–e^x sin x + e^x cos x]

`f^(''')(x)|_(x = π/2) < 0` and `f^(''')(x)|_(x = (3π)/2) > 0`

∴ SIope is maximum at x = `π/2`.