How can I find the absolute maximum value of the function given that both $a$ and $b$ are both positive constants

Question

I am given this function: $f(x)= x^a(1-x)^b$ and I am told that I have to find the absolute maximum value of this function within the interval of $0 \leq x \leq 1.$ Assume that both $a$ and $b$ are positive.

Here is my reasoning so far:

Using the Extreme Value Theorem, if $f$ is closed on $[a,b]$, then there is an absolute maximum value $f(c)$ or an absolute minimum value $f(d)$at some numbers $c$ and $d$ in $[a,b]$ .

Now, I have plugged both $f(0)$ and $f(1)$ inside my function, in which $f(0) = f(1)= 0$.

Since $a$ and $b$ are positive, there is an absolute max and definitely no absolute min.

Therefore my question is: How should I solve the absolute maximum? Should I derive from there? If I derive it, it will get somewhat messy.. is there an alternative (and more efficient way) of approaching this problem?

Thanks!

Answer 1

Hint: $f$ attains its maximum exactly when $\ln f(x)=a \ln x +b \ln (1-x)$ attains its maximum. It is easy to differentiate this an see when the derivative is $0$. You can also compute the second derivative to see if there is a maximum or a minimum at the point(s) where the first derivative is $0$.