If $x$ is real find the maximum possible value of $10^x-100^x$ According to the person who gave this question it apparently has something to do with the range of a quadratic expression. But I can't see the connection with a quadratic equation.
So I tried to solve this by finding the maxima of the expression.
But I don't know how to do it as it's an exponential function.
All I can infer from this is that $x$ must be negative.

$$F(x)=10^x-100^x=10^x(1-10^x)$$
Let $$f(a)=a(1-a)$$
$$f'(a)=1-2a$$
the maximum of $ f(a)  $ is $$ f(\frac 12)=\frac 14.$$
Thus, the maximum of $ F(x) $ is $ \frac 14 $ attained for $ x$ such that
$$10^x=\frac 12 = e^{x\ln(10)}$$
