Question

Verify Rolle’s Theorem for the function f(x) = e^x (sin x – cos x) on `[ (π)/(4), (5π)/(4)]`.

Answer 1

f (x) = e^x (sin x - cos x), `[ (π)/(4), (5π)/(4)]`

Differentiate w.r.t. x, we have

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

f (x) = e^x [cos x + sin x + sin x - cos x]

f (x) = 2 e^x sin x

For maxima or minima, we have

f'(x) = 0

2e^x sin x = 0

sin x = 0

x = n π

Thus,    x = π, ∵ lies between `[ (π)/(4), (5π)/(4)]`

So, Rolle's theorem is verified.