Question

Verify Mean Value Theorem, if f(x)=x^2-4x-3 in the interval [a,b], wherea=1 and b=4.

Answer 1

◆ function f(x) is polynomial so it is differentiable as well as continuous.

so

◆Check for f(a)=f(b)

$f'(c) = \frac{f(4) - f(1)}{4 - 1} \\ f'(c) = \frac{3}{3} = 1$
f'(c)=2c-4
2c-4=1
c=5/2

Answer 2

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 
$\bf{f'(c)=\frac{f(b)-f(a)}{b-a}}$

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).