Find all value of a

Question

For $a\geq 0$ . It know that inequality $3^x+a^x\ge 6^x+9^x$ Is true with $\forall x\in R$ find all value of a .

I use derivative but not know how must go on

Answer 1

Consider $f(x)=3^x+a^x-6^x-9^x$ we have that $f(0)=0$. If the minimum value of the function occurs at $x\neq0$ then the value of derivative at $f'(0)\neq 0$. Why? $x\rightarrow -\infty$ we have that $f'(x)=0$ and at $x\rightarrow\infty$ we have $f'(x)\rightarrow\infty$. So by intermediate value theorem the value of derivative will be non-zero at $0$ if derivative is $0$ at some point less than or more than $0$. This would imply that $f(x)$ will change signs at $0$.

Therefore $f'(x)=3^x\ln3+a^x\ln a-6^x\ln6-9^x\ln 9$ at $x=0$ becomes $$\ln3+\ln a-\ln 6-\ln9=0 \implies a=18$$