How can I find the approximate locations of solutions of $a^x\ln(a)+b^x\ln(b)+c^x\ln(c)=0$ provided that $a+b+c=3$ and $a,b,c$ are not all equal?

Question

I need to sketch a rough graph of $f(x)=a^x+b^x+c^x$ provided that $a+b+c=3$ and $a,b,c$ are not all equal. $$f'(x)=a^x\ln(a)+b^x\ln(b)+c^x\ln(c)$$

How can I find the approximate locations of solutions of $$a^x\ln(a)+b^x\ln(b)+c^x\ln(c)=0$$ I would then use this to find the maxima,minima and points of inflections of the curve.

I took the function $f(x)=a^x+b^x+c^x$

I figured there must be one solution in $x∈(0,1)$ by Rolle's theorem because $f(0)=f(1)=3$ and the function is continuous.

I don't know how to find any other solutions if any. By using a graphical calculator to sketch some curves, it seems like there are no other such points. If this is true how can I prove it?

Thanks in advance!

Regards

Answer 1

Taking the next derivative, $$f''(x) = a^x(\ln a)^2 + b^x(\ln b)^2 + c^x(\ln c)^2 > 0.$$ Therefore there are no points of inflexion, and there is a unique minimum for some $x \in (0,1)$. Indeed, experimenting with a graphing calculator shows that the minimum is almost always close to $1/2$.