Question

Verify Mean Value Theorem, if f (x) = x² – 4x – 3 in the interval [a, b], where a = 1 and b =4.

Answer 1

Answer:

According to mean values theorem,

for a function f : [a, b] → R , if

(a)f is continuous on [a, b]

(b)f is differentiable on (a, b)

Then there exists some c ∈ (a, b) such that 

f'(c) =f(b)-f(a)/b-a

given, f(x) = x² - 4x - 3

as we know, polynomial function is continuous function. so, f is continuous.

and, f'(x) = 2x - 4 , it is also differentiable.

now, Let c is the point in [1, 4]

so,f'(c) =2c - 4 = {f(4) - f(1)}/(4 - 1)

f(4) = 4² - 4(4) - 3 = 16 - 16 - 3 = -3

f(1) = 1² - 4(1) - 3 = 1 - 4 - 3 = -6

hence, f'(c) = 2c - 4 = {-3 + 6}/3

=> 2c - 4 = 1

=> c = 5/2

The Mean Value Theorem is verified for the given f(x).

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