if $f$ is differentiable over an interval $I$ and $f'$ is decreasing over $I,$ then $f$ is concave down over $I$

if $f$ is differentiable over an interval $I$ and $f'$ is increasing over $I,$ then $f$ is concave up over $I$

the upward or downward curve of the graph of a function

suppose $f$ is twice differentiable over an interval $I;$ if $f'' > 0$ over $I,$ then $f$ is concave up over $I;$ if $f'' < 0$ over $I,$ then $f$ is concave down over $I$

first derivative test

let $f$ be a continuous function over an interval $I$ containing a critical point $c$ such that $f$ is differentiable over $I$ except possibly at $c;$ if $f'$ changes sign from positive to negative as $x$ increases through $c,$ then $f$ has a local maximum at $c;$ if $f'$ changes sign from negative to positive as $x$ increases through $c,$ then $f$ has a local minimum at $c;$ if $f'$ does not change sign as $x$ increases through $c,$ then $f$ does not have a local extremum at $c$

if $f$ is continuous at $c$ and $f$ changes concavity at $c,$ the point $(c, f (c))$ is an inflection point of $f$

second derivative test

suppose $f' (c) = 0$ and $f''$ is continuous over an interval containing $c;$ if $f'' (c) > 0,$ then $f$ has a local minimum at $c;$ if $f'' (c) < 0,$ then $f$ has a local maximum at $c;$ if $f'' (c) = 0,$ then the test is inconclusive

4.5 Derivatives and the Shape of a Graph: practice

For the function $y = x^{3},$ is $x = 0$ both an inflection point and a local maximum/minimum?

Is it possible for a point $c$ to be both an inflection point and a local extremum of a twice differentiable function?

Explain whether a concave-down function has to cross $y = 0$ for some value of $x .$

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−2, 0). Then it continues increasing a little before decreasing and crossing the x axis at (−1, 0). It achieves a local minimum at (1, −6) before increasing and crossing the x axis at (2, 0).

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts negative and touches the x axis at the origin. Then it decreases a little before increasing to cross the x axis at (1, 0) and continuing to increase.

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts at (−2, 0), decreases to (−1.5, −1.5), increases to (−1, 0), and continues increasing before decreasing to the origin. Then the other side is symmetric: that is, the function increases and then decreases to pass through (1, 0). It continues decreasing to (1.5, −1.5), and then increase to (2, 0).

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where

$f$ is increasing and decreasing and
the minima and maxima are located.

The function f’(x) is graphed. The function starts at (−2, 0), increases and then decreases to (−1, 0), decreases and then increases to an inflection point at the origin. Then the function increases and decreases to cross (1, 0). It continues decreasing and then increases to (2, 0).

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where

$f$ is increasing and decreasing and
the minima and maxima are located.

The function f’(x) is graphed. The function starts negative and crosses the x axis at the origin, which is an inflection point. Then it continues increasing.

For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals where $f$ is concave up and concave down.

The function f’(x) is graphed. The function is linear and starts negative. It crosses the x axis at the origin.

For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals where $f$ is concave up and concave down.

The function f’(x) is graphed. The function resembles the graph of x3: that is, it starts negative and crosses the x axis at the origin. Then it continues increasing.

For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals where $f$ is concave up and concave down.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing to a local maximum at (0, 1), at which point it decreases and touches the x axis at (1, 0). It then increases.

For the following exercises, draw a graph that satisfies the given specifications for the domain $x ϵ [- 3, 3] .$ The function does not have to be continuous or differentiable.

$f^{'} (x) > 0$ over $x > 2, - 3 < x < - 1, f^{'} (x) < 0$ over $- 1 < x < 2, f^{''} (x) < 0$ for all $x$

For the following exercises, draw a graph that satisfies the given specifications for the domain $x ϵ [- 3, 3] .$ The function does not have to be continuous or differentiable.

There is a local maximum at $x = 2,$ local minimum at $x = 1,$ and the graph is neither concave up nor concave down.