Find the relative extrema

Question

Find the relative extrema for f(x)=2sin² x-cos2x on the interval [0,2π]. Also find the x coordinates of the inflection points of f(x) on this interval. Give the intervals where the function is increasing, where it is decreasing.

Answer 1

To find the relative extrema, we use the first derivative test.  This is set the derivative of f(x) equal to zero.  This is because the derivative is the slope of the tangent line, and the line that tangent to any local extrema has a slope of zero.

 

 

f'(x) = 0

 

2*2sin(x)cos(x) + 2sin(2x) = 0

 

 

Using the double angle identity:  sin(2x) = 2sin(x)cos(x)

 

2sin(2x) + 2sin(2x) = 0

 

4sin(2x) = 0

 

  sin(2x) = 0

 

 

Use the substitution          q = 2x

 

 

sin(q) = 0

 

q = 0

q = π

q = 2π

 

 

0 = 2x               π = 2x                2π = 2x

 

0 = x                 π/2 = x                π = x

 

 

These values of x are our critical points.  They are the location of the local extremas.

 

Next, we use test points to evaluate the derivative.  Lets use  x=π/3 , x=3π/4 , and  x=7π/4.

 

 

Evaluate

 

f'(π/3)  ,   f'(3π/4) , and   f'(7π/4)

 

 

Also evaluate f(x) at the interval endpoints.

 

f(0)   and   f(2π)          for accuracy.

 

 

The 3 critical points are only relative extremas if the derivative before the point and the derivative after the point changes signs.  If the derivative goes from negative to positive, then the critical point is a minimum.  If the derivative goes from positive to negative, then the critical point is a maximum.

 

You can also get your intervals of increasing and decreasing from the local extremas.

 

Once you have your minimum and maximum, evaluate f(x) at those critical points to get your local extremas.

 

 

 

To find the inflection points, set the second derivative equal to zero.  This is called the second derivative test.  To find the second derivative, derive the first derivative.

 

 

f''(x) = 0

 

8cos(2x) = 0

 

cos(2x) = 0

 

 

Use the substitution       r = 2x

 

cos(r) = 0

 

r = π/2

r = 3π/2

 

 

π/2 = 2x                  3π/2 = 2x

 

π/4 = x                    3π/4 = x

 

These are your possible inflection points.  Inflection points show where f(x) changes concavity.  Use tests points (just like in the first derivative test) and evaluate the second derivative using those test points.  If f''(x) changes signs before and after the possible points of inflection, then they are inflection points.