Let $f(x)=x^4+x^3-x-1$. Use Rolle's theorem to prove $4x^3+3x^2-1=0$ has at least one real root in [-1,1].
Do I have to let $f(x)=\sin(x)$ and continue to do it? I have no idea how to use the Rolle's theorem as this can be easily mixed up with LaGrange theorem.
Best Answer
Let $$f(x)=x^4+x^3-x-1$$ $$f(1)=0$$ $$f(-1)=0$$
Since we have $f(1)=f(-1)=0$
Then Using Rolle's Theorem
We can say that $\exists c\in[-1,1]$ such that
$$f'(c)=0$$ $$\implies f'(x)=g(x)=4x^3+3x^2-1 $$ has atleast one real root in $[-1,1]$