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^(3)-5x^(2)-3x, " " x in [1, 3]`
f(x) is a polynomial function.
` :. `f(x) is continuous in [1, 3] and differentiable in (1, 3).
`implies` f(x) satisfies all conditions of Lagrange's theorem in [1, 3].
` :. f'(x)=3x^(2)-10x -3 `
Let `f'(c)=(f(3)-f(1))/(3-1)`
`implies3c^(2)-10c -3=((3^(3)-5xx3^(3)-3xx3)-(1^(3)-5xx1^(2)-3xx1))/(2) `
`=((-27)-(-7))/(2)= -10`
` implies 3c^(2)-10c +7=0`
`implies (3c-7)(c-1)=0`
`impliesc=(7)/(3)" or "c=1`
` :. c=(7)/(3) in (1, 3)`
Hence, Lagrange's theorem is verified.
Again let `f'(c) = 0 `
`implies 3c^(2)-10c-3=0`
`implies c=(10+-sqrt(100+36))/(2xx3)=(10+-2sqrt(34))/(6)`
`impliesc= (5+-sqrt(34))/(3)in (1,3)`
` :. ` No value of c lies in (1, 3) for which f'(c) = 0.