Increase Profit sale car with respect to rebate Tesla makes $ 1,500 $ dollars on sale of model 3. It is estimated for every $100$ dollars of rebate, sales increase by $15$%. Use one variable optimization to maximize profits. * * * I did pass my calc I class so I know to do $y'(x)=0$ is where max/min are depending on concavity of function. I am just having trouble getting the right equation from the words. I think I am missing something * Base income 1,500 dollars * Rebate is 100$ * sales increase 15% called $x(\phi)=(1+.15 \phi)$ that is increase of sales and profit/car $p=1,500-100 \phi$. so $$P=p*x(\phi)=(1,500-100\phi)(1+.15\phi)$$

$P'(\phi)=125-30\phi=0 \quad \Rightarrow \phi=25/6$

Since $P'(\phi)>0$ for $\phi<25/6$ and $P'(\phi)<0$ for $\phi>25/6$

We have the maximum at $\phi=25/6$

But I assume that we cannot have a partial number of rebates so since $25/6$ is closer to $4$ than $5$, the optimal number is \$ $400$ of rebate.

Sanity Check: $P(4)=(1500-400)(1+.6)=1760$

Note: I have only assumed the information that was given and have used your model.