The Second Derivative Test

Concavity and Inflection Points

Example:

Consider a typical ski slope.  At what point on the slope are you having the most fun?  In other words where is the slope the greatest? Definition If f '(x) is increasing, then the function is concave up and if f '(x) is decreasing then the function is concave down.  To determine whether the derivative is increasing, we take the second derivative.

Example:

Let

f(x)  =  3x2 - x3

Then

f '(x)  =  6x - 3x2

and

f ''(x)  =  6 - 6x.

Solving

6 - 6x  >  0

we see that the function is concave up when x <1.

Solving

6 - 6x  <  0

we see that the function is concave down when x > 1.

When x = 1 we say that f(x) has an inflection point. Exercise:
Determine where the function is concave up and concave down:

1. f(x)  =  x3 - x

2. f(x)  =  x4 - 6x2

3. The Second Derivative Test

Suppose that a twice differentiable function has a relative maximum at x = c.  The by the first derivative test, the derivative is positive to the left of c and negative to the right of c.  Going from positive to negative means the derivative is decreasing at x = c.  A decreasing first derivative implies a negative second derivative.  Similarly at a relative minimum the second derivative is positive.  This implies

 Let f be a twice differentiable function near c such that f '(c) = 0.  Then If f ''(x) > 0 then f(c) is a relative minimum. If f ''(x) < 0 then f(c) is a relative maximum. If f ''(x) = 0 then use the first derivative test.

Example:

Let

f(x) = x4 - 4x3

Then

f '(x) =  4x3 - 12x2

which is zero at x = 0 and x = 3

f ''(x)  =  12x2 - 24x

f ''(0)  =  0

and

f ''(3)  >  0

Hence at x = 3 there is a relative minimum.  At x = 0 we use the first derivative test.  To the left of 0,

f '(x) < 0

and to the right of 0

f '(x) <0

thus at x = 0 there is no local extrema.

Exercises:

For the following determine the concavity, inflection points, relative extrema and intervals where the function is increasing and intervals where the function is decreasing.

1. f(x) = 5 + 3x2 - x3

2. f(x)  =  x4 - 4x3 + 2

3. f(x) =   x + 4/x