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.

$10^x - 100^x = 10^x -(10^2)^x = 10^x - (10^x)^2$
Let $y = 10^x$ then $10^x - 100^x = y - y^2$.
To find the maximum of that... well, use calculus.  $f(y) = y-y^2$ so $f'(y) = 1-2y=0 \implies y=\frac 12$ so the maximum value is $y = \frac 12$.
So $y = 10^x = \frac 12$
And $x =\log_{10} \frac 12 = -\log_{10} 2$.
