4.5 Derivatives and the shape of a graph  (Page 6/14)

 Page 6 / 14

Consider the function $f\left(x\right)={x}^{3}-\left(\frac{3}{2}\right){x}^{2}-18x.$ The points $c=3,-2$ satisfy ${f}^{\prime }\left(c\right)=0.$ Use the second derivative test to determine whether $f$ has a local maximum or local minimum at those points.

$f$ has a local maximum at $-2$ and a local minimum at $3.$

We have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. In the next section we discuss what happens to a function as $x\to \text{±}\infty .$ At that point, we have enough tools to provide accurate graphs of a large variety of functions.

Key concepts

• If $c$ is a critical point of $f$ and ${f}^{\prime }\left(x\right)>0$ for $x and ${f}^{\prime }\left(x\right)<0$ for $x>c,$ then $f$ has a local maximum at $c.$
• If $c$ is a critical point of $f$ and ${f}^{\prime }\left(x\right)<0$ for $x and ${f}^{\prime }\left(x\right)>0$ for $x>c,$ then $f$ has a local minimum at $c.$
• If $f\text{″}\left(x\right)>0$ over an interval $I,$ then $f$ is concave up over $I.$
• If $f\text{″}\left(x\right)<0$ over an interval $I,$ then $f$ is concave down over $I.$
• If ${f}^{\prime }\left(c\right)=0$ and $f\text{″}\left(c\right)>0,$ then $f$ has a local minimum at $c.$
• If ${f}^{\prime }\left(c\right)=0$ and $f\text{″}\left(c\right)<0,$ then $f$ has a local maximum at $c.$
• If ${f}^{\prime }\left(c\right)=0$ and $f\text{″}\left(c\right)=0,$ then evaluate ${f}^{\prime }\left(x\right)$ at a test point $x$ to the left of $c$ and a test point $x$ to the right of $c,$ to determine whether $f$ has a local extremum at $c.$

If $c$ is a critical point of $f\left(x\right),$ when is there no local maximum or minimum at $c?$ Explain.

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

It is not a local maximum/minimum because ${f}^{\prime }$ does not change sign

For the function $y={x}^{3},$ is $x=0$ an inflection point?

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

No

Why do you need continuity for the first derivative test? Come up with an example.

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

False; for example, $y=\sqrt{x}.$

Explain whether a polynomial of degree $2$ can have an inflection point.

For the following exercises, analyze the graphs of ${f}^{\prime },$ then list all intervals where $f$ is increasing or decreasing. Increasing for $-2 and $x>2;$ decreasing for $x<-2$ and $-1  Decreasing for $x<1,$ increasing for $x>1$  Decreasing for $-2 and $1 increasing for $-1 and $x<-2$ and $x>2$

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

1. $f$ is increasing and decreasing and
2. the minima and maxima are located.  a. Increasing over $-22,$ decreasing over $x<-2,$ $-1 b. maxima at $x=-1$ and $x=1,$ minima at $x=-2$ and $x=0$ and $x=2$  a. Increasing over $x>0,$ decreasing over $x<0;$ b. Minimum at $x=0$ For the following exercises, analyze the graphs of ${f}^{\prime },$ then list all inflection points and intervals $f$ that are concave up and concave down. Concave up on all $x,$ no inflection points  Concave up on all $x,$ no inflection points  Concave up for $x<0$ and $x>1,$ concave down for $0 inflection points at $x=0$ and $x=1$

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

$f\left(x\right)>0,{f}^{\prime }\left(x\right)>0$ over $x>1,-3 over $0

${f}^{\prime }\left(x\right)>0$ over $x>2,-3 over $-1 for all $x$

$f\text{″}\left(x\right)<0$ over $-10,-3 local maximum at $x=0,$ local minima at $x=\text{±}2$

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

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