Find maximum decay rate, given $y(x)=5xe^{-2x} \text{kg/year} $

Question

Let the Decay rate of a metal is $$ y(x)=5xe^{-2x} \text{kg/year} $$

Find the maximum Decay rate .

Solution :

The decay rate is given by

$$ y(x)=5x e^{-2x} $$

For maximum decay rate

$$ y'(x)=0 \\ \implies 5e^{-2x}-10 x e^{-2x}= 0 \\ \implies 5-10x=0 \\ \implies x=\frac{1}{2} \text{ year}$$

Am I right?

Answer 1

• Remember to check the second derivative to make sure that it is indeed the maximum.

• If the maximum is indeed attained at $x=\frac12$, you want to evaluate $y(0.5)$.

• Your working is correct so far. Just that it is not complete.