Question

f(x) = e^-x sin x rolles theorem​

Answer 1

We have , f(x) = e2x(sin2x – cos 2x) x ε [π/8, 5π/8]

(1) As sine, cosine and exponential function are always continuous, hence given function f(x) is continuous in [π/8, 5π/8] .

(2) f’(x) = e2x × 2 (sin 2x – cos 2x) + e2x (2 cos 2x + 2 sin 2x)

= 2 e2x (sin 2x – cos 2x + cos 2x + sin 2x)

= 2 e2x(2 sin 2x) = 4 e2x sin 2x.

Thus derivatives exists in the given interval and function is differentiable.

(3) f(π/8) = eπ/4 (sin π/4 – cos π/4) = eπ/4 ×0 = 0 .

f(5π/8) = e5π/4 (sin 5π/4 – cos 5π/4) = e5π/4 × 0 = 0 .

Therefore , f(π/8) = f(5π/8)

Now f’(c) = 0

Or, 4 e2c sin 2c = 0

Or, sin 2c = 0 [As e2c ≠ 0]

Hence, 2c = 0 , π , 2π , 3π …………. .

Or, c = 0 , π/2 , π , 3π/2 ………… .

Therefore , π/2 ε (π/8 , 5π/8) .

Hence Rolle’s theorem is verified.