Question

Verify Lagranage's mean value theorem for the $f\left(x\right)=x^3-2x^2-x+3$ in $[0,1]$

Answer 1

f(x) is a polynomial function.

It is known to us that polynomial function is always continuous and differentiable.

therefore, f(x) is continuous in [0,1]

and f(x) is differentiable in (0,1)

By Lagrange's mean value theorem,

$f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\Rightarrow f^{\prime }\left(c\right)=\frac{f\left(1\right)-f\left(0\right)}{1-0}\Rightarrow 3c^2-4c-1=(1-3)\Rightarrow 3c^2-4c+1=0\left(bymiddletermsplitting\right)\Rightarrow c=\frac{1}{3};1$