1
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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 $[-1,1]$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.