Determine whether $e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$ is true or false?

Question

The statement is

$e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$

I think the above statement is true by calculating approximate value at some points.

But how to prove it properly?

Answer 1

Let $y=e^x$. Then $y\in (0,1)$

Can you show that for all $y\in (0,1)$, $y(1-y)\leq\frac{1}{4}$? This is just a second degree polynomial.

Answer 2

Note that $e^x\in(0,1)$ for $x<0$.

Note $y(1-y)$ is a '$\bigcap$' quadratic with roots at zero and one and so by symmetry has a maximum at the midpoint $y=1/2$ where it equals $1/4$.

Restricting to $y=e^x\in (0,1)$ gives $e^x(1-e^x)\in(0,1/4]$.

Answer 3

Just write: $$\forall x\in\mathbb{R},\ \mathrm{e}^x(1-\mathrm{e}^x)=-\left(\mathrm{e}^x-\frac12\right)^2+\frac14.$$ Then use the fact that the square of a real number is non-negative to obtain: $$\forall x\in\mathbb{R},\ \mathrm{e}^x(1-\mathrm{e}^x)\leq\frac14.$$ (So the inequality is valid, even for non-negative $x$'s). You also get, for free, that the equality occurs exactly once, for $x=-\ln(2)$.