Verify Rolle's theorem for the following function :`f(x)=x^2-4x+10` on `[0, 4]`

Explanation

Given, ` f(x) = x^(2) - 4x + 10`
` :' ` f(x) is a polynomial,
(a) It is continuous on (0,4)
(b) It is differentiable on (0,4)
(c ) `f(0) = 0^(2) - 4 * 0 + 10 = 10`
` f(4) = 4^(2) - 4 * 4 + 10 = 10`
` f(0) = f(4) = 10`
Thus, f (x) satisfies all conditions of Rolle's theorem.
Thus, f' (x) will vanish for at least one point C in (0, 4).
Taking f'(C) = 0 and f'(C) = 2C - 4
` :." " 2C - 4 = 0`
` :. " " C = 2 in (0,4)`
Thus, Rolle's theorem is veridied .