Finding Minimum and Maximum Derivatives

• 1
  Differentiation is a valuable tool for investigating the behavior of a function.graph image by Attila Toro from Fotolia.com
  
  Find the derivative of your function.
  
  Some examples:
  
  If your function, f(x)=3x, then your derivative, f'(x)=3.
  
  If g(y)=4(y-2)^2 + 6, then your derivative, g'(y)=8*(y-2).
  
  If h(z)=sin(z), then h'(z)=cos(z).
  
  
• 2). Find the derivative of the derivative of your function, otherwise known as the second derivative.
  
  From the examples:
  
  For f(x)=3x, and f'(x)=3, then f''(x)=0.
  
  For g(y)=4(y-2)^2 + 6, and g'(y)=8*(y-2), then g''(y)=8.
  
  For h(z)=sin(z), and h'(z)=cos(z), then h''(z)=-sin(z).
  
  
• 3
  Straightforward steps will help you identify maxima and minima.calculator image by Randy McKown from Fotolia.com
  
  Set the second derivative equal to zero. The second derivative of your function will be equal to zero only when the first derivative has a minimum or maximum.
  
  Each of the three examples above demonstrates different behavior. For f(x)=3x, f''(x)=0. For what values of x is f''=0? All of them. Therefore, your derivative has a minimum or maximum at every point, which doesn't make sense until you remember that the derivative, f'(x) is equal to 3 everywhere. So it has no minima or maxima, or it has the same maximum and minimum everywhere, which is 3.
  
  For g(y)=4(y-2)^2 + 6, g''(y)=8. For what values of y is g''=0? None of them; it's always equal to 8, so the derivative of your function has no minima or maxima. Again, it seems strange until you look at the graph and see that your initial quadratic function g(y) has a first derivative that's just a straight line---no dips or bumps to make extrema.
  
  For h(z)=sin(z), h''(z)=-sin(z). For what values of z is -sin(z)=0? At z=0, +/-pi, +/-2*pi, etc. Now look back at the first derivative and plug in the values of z that we now believe to correspond to minima and maxima. h'(z)=cos(z). Cos(0)=1, which we know is a maximum for the cosine function. Cos(pi)=-1, which we know is a minimum for cosine, etc.
  
  
• 4). Now restrict the range for your independent variable to find the relative maximum and minimum derivatives. In this context, relative maximum just means the maximum over a given range of independent variables. In our third example above, we could ask for the relative maximum between z= 3*pi and 5*pi, and we'd find extrema at 3*pi, 4*pi, and 5*pi. For this example, the cosine function is well known to the point where we know that it's minimum at 3 pi and 5 pi, and maximum at 4 pi.
  
  This step has given us the extrema, but it doesn't tell us for sure which are maxima and which are minima. One final step will clear up the remaining confusion.
  
  
• 5). Take the derivative of your function one more time. If it's positive at the extremum, then it's at a minimum, if it's negative, you're at a maximum.
  
  Our example again: the second derivative is h''(z)=-sin(z), the derivative of that is h'''(z)=-cos(z). In the range z=3*pi to 5*pi, the second derivative was equal to zero at 3*pi, 4*pi, and 5*pi, so those are the values we're interested in. -cos(3*pi)=1, which is positive, so the extrema we found is a minimum. -cos(4*pi)=-1, so the extrema is a maximum. And cos(5*pi)=1, so the extrema there is another minimum. All that is consistent with what we know of the cosine function.