Answer to Question #257102 in Calculus for Nazmul

Question

Verify mean value theorem, if f(x) =x²-4x-3 in the interval [a,b] where a= 1 and b= 4 x

Answer 1

$f(x)=x^2-4x-3$ is continuous on $[1,4]$ as polynomial.

$f(x)=x^2-4x-3$ is differentiable on $(1,4)$ as polynomial.

Therefore, by the Mean Value Theorem, there is a number $c$ in $(1, 4)$ such that

$$f(4)-f(1)=f'(c)(4-3)$$

$$f(4)=4^2-4(4)-3=-3$$

$$f(1)=1^2-4(1)-3=-6$$

$$f'(x)=2x-4$$

$$f'(c)=2c-4$$

Substitute

$$-3-(-6)=(2c-4)(4-1)$$

$$2c-4=1$$

$$c=\dfrac{5}{2}$$

$$1<\dfrac{5}{2}<4=>c\in (1, 4)$$

Hence the hypotheses of the Mean Value Theorem are satisfied.