Question

lf Rolle's theorem for f(x) = e^x (sin x – cos x) is verified on `[π/4, (5π)/4]`, then the value of c is ______.

विकल्प

• `π/3`

• `π/2`

• `(3π)/3`

• π

Answer 1

lf Rolle's theorem for f(x) = e^x (sin x – cos x) is verified on `[π/4, (5π)/4]`, then the value of c is `underlinebb(π/2)`.

Explanation:

Given, f(x) = e^x (sin x – cos x)

`\implies f^'(x) = e^x d/dx (sinx - cosx) + (sinx - cosx) d/dx (e^x)`

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

= 2e^x sin x

We know that, if Rolle's theorem is verified, then their exist `c ∈ (π/4, (5π)/4)`, such that f'(c) = 0

∴ 2e^c sin c = 0

`\implies` sin c = 0

`\implies` c = `π/2 ∈ (π/4, (5π)/4)`