Question

for the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down; and e) sketch the graph. k(x)=4x^4 + 16x^3

Explanation:

Step1: Find the first - derivative

Differentiate $k(x)=4x^{4}+16x^{3}$ using the power rule. The power rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$.
$k^\prime(x)=16x^{3}+48x^{2}=16x^{2}(x + 3)$

Step2: Find critical points

Set $k^\prime(x)=0$.
$16x^{2}(x + 3)=0$
$x = 0$ or $x=-3$
When $x=-3$, $k(-3)=4(-3)^{4}+16(-3)^{3}=4\times81-16\times27=324 - 432=-108$
When $x = 0$, $k(0)=0$
So the critical points are $(-3,-108)$ and $(0,0)$

Step3: Classify critical points

Use the first - derivative test.
Choose test points: Let's choose $x=-4$, $x=-1$ and $x = 1$
$k^\prime(-4)=16(-4)^{2}(-4 + 3)=16\times16\times(-1)<0$
$k^\prime(-1)=16(-1)^{2}(-1 + 3)=16\times1\times2>0$
$k^\prime(1)=16\times1^{2}(1 + 3)=16\times4>0$
Since $k^\prime(x)$ changes sign from negative to positive at $x=-3$, $(-3,-108)$ is a relative minimum. Since $k^\prime(x)$ does not change sign at $x = 0$, $(0,0)$ is neither a relative maximum nor a relative minimum.

Step4: Find the second - derivative

Differentiate $k^\prime(x)=16x^{3}+48x^{2}$
$k^{\prime\prime}(x)=48x^{2}+96x=48x(x + 2)$

Step5: Find points of inflection

Set $k^{\prime\prime}(x)=0$
$48x(x + 2)=0$
$x = 0$ or $x=-2$
When $x=-2$, $k(-2)=4(-2)^{4}+16(-2)^{3}=4\times16-16\times8=64 - 128=-64$
So the points of inflection are $(0,0)$ and $(-2,-64)$

Step6: Find intervals of concavity

Choose test points: Let's choose $x=-3$, $x=-1$ and $x = 1$
$k^{\prime\prime}(-3)=48(-3)(-3 + 2)=48\times(-3)\times(-1)>0$
$k^{\prime\prime}(-1)=48(-1)(-1 + 2)=48\times(-1)\times1<0$
$k^{\prime\prime}(1)=48\times1\times(1 + 2)=48\times3>0$
The function is concave up on $(-\infty,-2)\cup(0,\infty)$ and concave down on $(-2,0)$

Step7: Find intervals of increase and decrease

The function is decreasing on $(-\infty,-3)$ and increasing on $(-3,\infty)$

Answer:

a) Critical points: $(-3,-108)$ (relative minimum), $(0,0)$ (neither)
b) Increasing on $(-3,\infty)$, decreasing on $(-\infty,-3)$
c) Points of inflection: $(0,0)$ and $(-2,-64)$
d) Concave up on $(-\infty,-2)\cup(0,\infty)$, concave down on $(-2,0)$
e) To sketch the graph: Mark the critical points $(-3,-108)$ and $(0,0)$, the points of inflection $(-2,-64)$ and $(0,0)$. The function is decreasing until $x=-3$, then increasing. It is concave up until $x=-2$, concave down from $x=-2$ to $x = 0$ and concave up again for $x>0$.