# 4.1 | Using First and Second Derivatives

Suppose $$f$$ is defined on an interval and has a local maximum or minimum at $$x = a,$$ which is not an endpoint of the interval. If $$f$$ is differentiable at $$x=a,$$ then $$f'(a) = 0.$$ Thus, $$a$$ is a critical point.
The First-Derivative Test for Local Maxima and Minima:
Suppose $$(p, f(p))$$ is a critical point of a continuous function $$f.$$ Moving from left to right:

• If $$f’$$ changes from negative to positive at $$p,$$ then $$f$$ has a local minimum at $$p.$$
• If $$f’$$ changes from positive to negative at $$p,$$ then $$f$$ has a local maximum at $$p.$$
The Second-Derivative Test for Local Maxima and Minima:

• If $$f'(p) = 0$$ and $$f^{\prime \prime }(p) > 0,$$ then $$f$$ has a local minimum at $$p.$$
• If $$f'(p) = 0$$ and $$f ^{\prime \prime } (p) < 0,$$ then $$f$$ has a local maximum at $$p.$$
• If $$f'(p) = 0$$ and $$f^{\prime \prime } (p) = 0,$$ then the test is inconclusive.
Suppose $$f^{\prime \prime }$$ is defined on both sides of $$p$$:

• If $$f^{\prime \prime }$$ is zero or undefined at $$p,$$ then $$(p,f(p))$$ is an inflection point.
• To test whether $$(p,f(p))$$ is an inflection point, check whether $$f^{\prime \prime }$$ changes sign at $$p.$$