Monotonicity, extrema, and concavity | Code Scholars Tutoring

Understand the idea

Use a sign chart of the first derivative to determine increase and decrease and to classify local extrema by sign changes. Use the second derivative or the monotonicity of the first derivative to determine concavity. Inflection points require a change in concavity at a point on the graph. A zero second derivative alone gives only a candidate; the second derivative test is inconclusive when that derivative is zero.

f'>0\Rightarrow f\text{ increasing};\quad f''>0\Rightarrow f\text{ concave up}

Worked example

Analyze local extrema and inflection for $\displaystyle f(x)=x^3-3x$ .

\displaystyle f'(x)=3(x-1)(x+1)

is positive outside

\displaystyle [-1,1]

and negative inside. There is a local maximum at

\displaystyle x=-1

and local minimum at

\displaystyle x=1

. Since

\displaystyle f''(x)=6x

changes from negative to positive at zero,

\displaystyle (0,0)

is an inflection point.

Try it yourself

Does $\displaystyle f''(0)=0$ prove an inflection for $\displaystyle f(x)=x^4$ ?

Show solution and reasoning

No. Here $\displaystyle f''(x)=12x^2$ is positive on both sides of zero. Concavity does not change.