Question

Verify Rolle's theorem for the given functions f (x) = sin x + cos x + 5, x ∈ [0, 2π]

Answer 1

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Answer 2

Answer:

We have given

f(x) = sinx + cosx + 5, x ∈ [0, 2π]

Now differentiate f(x) w. r. t. x

f'(x) = cosx - sinx

f(x) is differentiable on open interval(0, 2π) and continuous on closed interval[0, 2π]

Now, f(a) = f(0) = sin0 + cos0 + 5 = 1 + 5 = 6

f(b) = f(2π) = sin2π + cosπ + 5 = 0 + 1 + 5 = 6

f(a) = f(b) = 0

Thus, function satisfy all the condition of Rolle's Theorem.

Now we have to show to show that there exist some c∈(0, 2π) such that f'(c) = 0

f(x) = sinx + cosx + 5

Now differentiate f(x) w. r. t. x

f'(x) = cosx - sinx

f'(c) = cosc - sinc =0

cosc = sinc

$\dfrac {cosc}{sinc} = 1\\\\cotc = cot45^o\\c = 45^0$

c = π/4 ∈[0, 2π]

Hence, Rolle's theorem is verified.