Identifying Turning Points (Local Extrema) for a Function
Key Questions

Answer:
See below.
Explanation:
To find extreme values of a function
#f# , set#f'(x)=0# and solve. This gives you the xcoordinates of the extreme values/ local maxs and mins.For example. consider
#f(x)=x^26x+5# . To find the minimum value of#f# (we know it's minimum because the parabola opens upward), we set#f'(x)=2x6=0# Solving, we get#x=3# is the location of the minimum. To find the ycoordinate, we find#f(3)=4# . Therefore, the extreme minimum of#f# occurs at the point#(3,4)# . 
Any polynomial of degree
#n# can have a minimum of zero turning points and a maximum of#n1# . However, this depends on the kind of turning point.Sometimes, "turning point" is defined as "local maximum or minimum only". In this case:
 Polynomials of odd degree have an even number of turning points, with a minimum of 0 and a maximum of
#n1# .  Polynomials of even degree have an odd number of turning points, with a minimum of 1 and a maximum of
#n1# .
However, sometimes "turning point" can have its definition expanded to include "stationary points of inflexion". For an example of a stationary point of inflexion, look at the graph of
#y = x^3#  you'll note that at#x = 0# the graph changes from convex to concave, and the derivative at#x = 0# is also 0.If we go by the second definition, we need to change our rules slightly and say that:
 Polynomials of degree 1 have no turning points.
 Polynomials of odd degree (except for
#n = 1# ) have a minimum of 1 turning point and a maximum of#n1# .  Polynomials of even degree have a minimum of 1 turning point and a maximum of
#n1# .
So, in part, it depends on the definition of "turning point", but in general most people will go by the first definition.
 Polynomials of odd degree have an even number of turning points, with a minimum of 0 and a maximum of

For a differentiable function
#f(x)# , at its turning points,#f'# becomes zero, and#f'# changes its sign before and after the turning points.
I hope that this was helpful.
Questions
Graphing with the First Derivative

Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Identifying Stationary Points (Critical Points) for a Function

Identifying Turning Points (Local Extrema) for a Function

Classifying Critical Points and Extreme Values for a Function

Mean Value Theorem for Continuous Functions