Originally Posted by

**MarkFL** We are given:

$\displaystyle y=xe^{\Large{0.39\left(1-\frac{x}{1619}\right)}}-x$

I would choose to write this as:

$\displaystyle y=xe^{\Large{\frac{39}{161900}(1619-x)}}-x$

And so equating the derivative to zero, we obtain:

$\displaystyle \frac{dy}{dx}=e^{\Large{\frac{39}{161900}(1619-x)}}-\frac{39x}{161900}e^{\Large{\frac{39}{161900}(1619-x)}}-1=0$

Or:

$\displaystyle e^{\Large{\frac{39}{161900}(1619-x)}}(161900-39x)-161900=0$

This cannot be solved algebraically using elementary functions, so using a numeric root finding technique, we find:

$\displaystyle x\approx768.7939617019124301$

We see that at $\displaystyle x=\frac{161900}{39}$ the derivative is negative, and at $\displaystyle x=0$ the derivative is positive, and so by the first-derivative test, we may conclude that the extremum is a maximum.