Minimize Gallons per Mile

[OldMan]

At speed v, gallons/mile = \(\displaystyle av+\frac bv\). Problem is to minimize gallons/mile
I get the wrong answer.
Any assistance pointing out my error(s) will be appreciated.
Thank you


Minimize Gallons/Mile
\(\displaystyle \frac {d} {dx} \ av+\frac {b}{v}\)
\(\displaystyle = a - \frac {b} {v^2}\)
\(\displaystyle 0 = a - \frac {b} {v^2}\)
\(\displaystyle v=\sqrt \frac {b}{a}\)
The Book says: \(\displaystyle v=2\sqrt {ab}\)

 

[galactus]

Is that the problem in its entirety?. As is, you are correct in differentiating \(\displaystyle av+\frac{b}{v}\)

Let's work backwards from the book solution of \(\displaystyle 2\sqrt{ab}\)

That means the original should have been \(\displaystyle \frac{v^{3}}{12}-abv\)

If we differentiate this, we get \(\displaystyle \frac{v^{2}}{4}-ab\).

Setting this to 0 and solving for v gives \(\displaystyle v=\pm 2\sqrt{ab}\)