Question

Verify Rolle's theorem for the function  

f(x)=x²-5x+9 on [1,4]

Answer 1

The function f given as f(x)=x²-5x+9 is a polynomial function.
Hence

(i) it is continuous on [1,4]
(ii) differentiable on (1,4).

`Now, f(1)=1^2 - 5(1)+ 9 =1- 5+9 = 5`

`and f (4)= 4^2 - 5(4)+ 9 =16 - 20 + 9 = 5`

f (1)=f(4)
Thus, the function f satisfies all the conditions of the Rolle’s theorem.

therefore there exists c ∈ (1, 4) such that f '(c)= 0

`Now, f(x)=x^2-5x+9`

`therefore f'(x)=d/dx(x^2-5x+9)=2x-5xx1+0`

=2x-5

f'(c)=2c-5

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

`c=5/2 in (1,4)`

Hence, the Rolle’s theorem is verified