Verify Mean Value Theorem, if `f(x)=x^3-5x^2-3x`in the interval [a, b], where `a = 1 a n d b = 3`. Find all `c in (1, 3)`for which `f^(prime)(c)=0`.

Explanation

`f(x)=x^(2)-4x-3, " " x in [1, 4]`
f(x) is a polynomial function.
` :. ` f(x) is continuous in [1, 4] and differentiable in (1, 4).
Thus, f(x) satisfies all conditions of Lagrange's theorem is [1, 4].
` :. f'(x)=2x-4`
Let `f'(c)=(f(4)-f(1))/(4-1)`
`implies2c -4=((4^(2)-4xx4-3)-(1^(2)-4xx1-3))/(3) `
`=((-3)-(-6))/(3)=1`
` implies 2c=5`
`impliesc=(5)/(2) in (1,4)`
Hence, Lagrange's theorem is verified.