Question

Verify Rolle's theorem for the function $f\left(x\right)={\log }_e\left[\frac{x^2+ab}{x(a+b)}\right]+p$, for $[a,b]$ where $0<a<b$.

Answer 1

$f\left(x\right)={\log }_e\left[\frac{x^2+ab}{x(a+b)}\right]+p$, for $[a,b]$ where $0<a<b$

Now, $f\left(x\right)={\log }_{e}x$ functions are continuous as well as differentiable for $x>0$

Now, $f\left(x\right)={\log }_e\left[\frac{x^2+ab}{x(a+b)}\right]$ to be verifying Relle's theorem, must we have $\frac{x^2+ab}{x(a+b)}>0$

As $0<a<b$

$ab>0$

& $x^2>0$

Therefore $(x^2+ab)>0$

Now, $x(a+b)>0$, as x lies in $[a,b]$ and $(a+b)>0$

Thus $\frac{x^2+ab}{x(a+b)}>0$ and $f\left(x\right)$ follows Relle's theorem.