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4-3 Using Derivatives to Analyze Graphs of Functions

Learning Targets

You will be able to

Concepts / Definitions

The First Derivative Test

Suppose that $c$ is a critical number of a continuous function $f$.

  1. If $f’$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$.
  2. If $f’$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$.
  3. If $f’$ does not change sign at $c$, then $f$ has no local maximum or minimum at $c$.

Concavity

The graph of a differentiable function $y = f(x)$ is

  1. Concave up on an open interval $I$ if $y’$ is increasing on $I$
  2. Concave down on an open interval $I$ if $y’$ is decreasing on $I$

Concavity Test

[Suppose the graph $y = f(x)$ is twice differentiable]

  1. If $f’‘(x) > 0$ for all $x$ in some open interval $I$, then the graph of $f$ is concave upward on $I$
  2. If $f’‘(x) < 0$ for all $x$ in some open interval $I$, then the graph of $f$ is concave downward on $I$

Point of Inflection

A point where the graph of a function has a tangent line (is differentiable) and where the concavity changes is a point of inflection.

Concavity and Inflection

Second Derivative Test for Local Extrema

  1. If $f’(c) = 0$ and $f’‘(c) < 0$$, then $f$ has a local maximum at $x = c$.
  2. If $f’(c) = 0$ and $f’‘(c) > 0$$, then $f$ has a local minimum at $x = c$.

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