# Derivative question: Speed vs. km traveled

I'm copying my question below as it is:

At a speed of S km per hour, a car will travel y km o each liter of petrol, where

y= 5+(1/5)S-(1/800)S^2

Calculate the speed at which the car should be driven for maximum economy.

The answer is 80 km/h; but how to get there from here? Thanks for the help in advance.

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The fuel economy is given by kilometers travelled per liter of fuel, so it is given by $y$. Since you are looking for maximum economy, $y$ needs to be maximized. Now, note that $y$ is a function of $S$, so maximizing $y$ means finding a value $S_\max$ of $S$ such that $y(S_\max)$ (written as a function) becomes maximal.

Now, as you mention derivatives, you should know that if $y(S^*)$ is a maximum or minimum, $\frac{dy}{dS}(S^*) = 0$. So, you can find candidate values for $S_\max$ by finding the zeroes of the derivative. Of course, you will then need to show that this is actually a maximum, but this isn't hard...

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Thank you, i think what confused me a bit was the expression "maximum economy"; but now I get it. – Deniz Apr 19 '12 at 9:28

Hint

You need to maximize $y$ with respect to $S$, so ...

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step 1: find derivative and set to zero(find critical points) 2.use second derivative test or something like this to see that,critical point is maxsimum

so let see y'=1/5-s/400 y'=0 means that 1/5=s/400 s=400/5=80 so s=80,if we need show that s=80 is maxsmum use second derivative test,namely for maxsimum y'' have to be less then $0$

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