According to one model of bird flight, the power consumed by a pigeon flying at velocity v (in m/s) is P(v)=17v^-1 +10^-3*v^3 . Find the velocity that minimizes power consumption.

The function is obviously defined only for v gt 0 and is continuously differentiable on this interval. When v approaches zero the function tends to +oo, when v tends to +oo, the function also tends to +oo.
Thus it must have at least one local (and global) minimum and it is reached at the point(s) where P'(x) = 0.  Let's solve this equation:
P'(x) = -17 v^-2 + 3*10^-3*v^2 = 0.
This is equivalent to  17 v^-2 = 3*10^-3*v^2, or  v^4 = 17/3*10^3.
The only solution is  v = root(4)(17/3*10^3) approx  8.7 (m/s). This is the answer.
 

Take the derivative of P(v) .
P'(v)=-17v^-2+3*10^-3v^2
Set P'(v) equal to zero and solve for the critical values.
-17v^-2+3*10^-3v^2=0
-17+3*10^-3v^4=0
3*10^-3v^4=17
v^4=17/3*10^3
v^4=17/3*10^3
v=(17/3*10^3)^(1/4)~~8.676 m/s
We can check graphically that this is indeed a minimum.