Question

Verify Rolle's theorem for the given functions f (x) = x² - 5x + 9, x ∈ [1,4]

Answer 1

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Answer 2

Answer:

We have given

$f(x) = x^2 - 5x + 9$, x ∈ [1, 4]

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

f'(x) = 2x - 5

f(x) is differentiable on open interval(1, 4) and continuous on closed interval[1, 4]

Now,

$f(a) = f(1)= 1^2 - 5 \times 1 + 9 = 1 - 5 + 9 = 5\\f(b) = f(4) = 4^2 - 5 \times 4 + 9 = 16 -20 + 9 = 5$

f(a) = f(b) = 5

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

Now we have to show that there exist some c∈(1, 4) such that f'(c) = 0

$f(x) = x^2 - 5x + 9$

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

f'(x) = 2x - 5

f'(c) = 2c - 5 = 0

c = 5/2 = 2.5

c = 2.5 ∈[1, 4]

Rolle's theorem verified.