Verify Rolle's theorem for the following function `f(x) = x^2 - 5x + 9, x in [1,4]`

Explanation

Now `f(x)=x^(2)-5x+9` is a polynomial. Therefore it is continuous and differentiable in its domain `x in R.` Hence, it is continuous in the interval [1, 4] and differentiable in the interval (1, 4)
`f(1)=1-5+9=5`
`f(4)=16-20+9=5`
`f(1)=f(4)`
`therefore" "` Conditions of Roll's theorem are staified.
`f'(x)=2x-5`
`rArr" "f'(c)=2c-5`
`rArr" "f''(c)=0`.
`rArr" "2c-5=0, c =(5)/(2) int [1, 4]`
`f'(c)=0 " at " c=(5)/(2)`