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Problem: Find the relationship of $a$ and $b$ so that Rolles theorem applies for the function $f(x) = ax^2 + b(\ln x)$ on $[1,e]$. Find the value of $c$ for which it is verified.

Answer: the relationship between $a$ and $b$ is $b = a(1 - e^2)$ the value of $c$ is $c = \pm\frac 1 2\sqrt{e^2 - 1})$

I would like to double check my answer on this problem Any feedback would be appreciated. Thank you

Paul Enta
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pi314
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1 Answers1

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Since $f(1)=a$ and $f(e)=ae^2+b$, one condition is $f(1)=f(e)$, so $b=a(1-e^2)$.

The function is indeed differentiable on $[1,e]$, so for $b=a(1-e^2)$, Rolle's theorem applies.

Since $$ f'(x)=2ax+\frac{a(1-e^2)}{x} $$ you just need to find $x\in(1,e)$ such that $2ax^2+a(1-e^2)=0$.

If $a=0$, there is obviously no problem. If $a\ne0$…

egreg
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