Question

If f'(x) > 0 on an interval, then fis ------ on that interva
Increasing
Decreasing​

Answer 1

Answer:

Explain how the sign of the first derivative affects the shape of a function’s graph.

State the first derivative test for critical points.

Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph.

Explain the concavity test for a function over an open interval.

Explain the relationship between a function and its first and second derivatives.

State the second derivative test for local extrema.

Earlier in this chapter we stated that if a function  has a local extremum at a point  then  must be a critical point of  However, a function is not guaranteed to have a local extremum at a critical point. For example,  has a critical point at  since  is zero at  but  does not have a local extremum at  Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information

Step 1. The derivative is



The derivative  when  Therefore,  at  The derivative  is undefined at  Therefore, we have three critical points:   and  Consequently, divide the interval  into the smaller intervals  and 

Step 2: Since  is continuous over each subinterval, it suffices to choose a test point  in each of the intervals from step 1 and determine the sign of  at each of these points. The points  are test points for these intervals.

IntervalTest PointSign of  at Test PointConclusion is decreasing. is increasing. is increasing.