<< Chapter < Page | Chapter >> Page > |
Consider the function $f\left(x\right)={x}^{3}-\left(\frac{3}{2}\right){x}^{2}-18x.$ The points $c=3,\mathrm{-2}$ 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 $\mathrm{-2}$ 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 $x\to \text{\xb1}\infty .$ At that point, we have enough tools to provide accurate graphs of a large variety of functions.
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 $\mathrm{-2}<x<\mathrm{-1}$ and $x>2;$ decreasing for $x<\mathrm{-2}$ and $\mathrm{-1}<x<2$
Decreasing for $\mathrm{-2}<x<\mathrm{-1}$ and $1<x<2;$ increasing for $\mathrm{-1}<x<1$ and $x<\mathrm{-2}$ and $x>2$
For the following exercises, analyze the graphs of ${f}^{\prime},$ then list all intervals where
a. Increasing over $\mathrm{-2}<x<\mathrm{-1},0<x<1,x>2,$ decreasing over $x<\mathrm{-2},$ $\mathrm{-1}<x<0,1<x<2;$ b. maxima at $x=\mathrm{-1}$ and $x=1,$ minima at $x=\mathrm{-2}$ 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 for $x<0$ and $x>1,$ concave down for $0<x<1,$ inflection points at $x=0$ and $x=1$
For the following exercises, draw a graph that satisfies the given specifications for the domain $x=[\mathrm{-3},3].$ The function does not have to be continuous or differentiable.
$f\left(x\right)>0,{f}^{\prime}\left(x\right)>0$ over $x>1,\mathrm{-3}<x<0,{f}^{\prime}\left(x\right)=0$ over $0<x<1$
${f}^{\prime}\left(x\right)>0$ over $x>2,\mathrm{-3}<x<\mathrm{-1},{f}^{\prime}\left(x\right)<0$ over $\mathrm{-1}<x<2,f\text{\u2033}\left(x\right)<0$ for all $x$
Answers will vary
$f\text{\u2033}\left(x\right)<0$ over $\mathrm{-1}<x<1,f\text{\u2033}\left(x\right)>0,\mathrm{-3}<x<\mathrm{-1},1<x<3,$ local maximum at $x=0,$ local minima at $x=\text{\xb1}2$
There is a local maximum at $x=2,$ local minimum at $x=1,$ and the graph is neither concave up nor concave down.
Answers will vary
Notification Switch
Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?