Question

For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at ±∞. Then interpret this information to sketch a labeled graph of f.

$f\left(x\right)=x^3-2x^2-4x+8$

Short answer

sign charts forf, f',and f ''are following.

Graph of function is following.

Step-by-step solution

Step 1. Given information. 

The given function is$f\left(x\right)=x^3-2x^2-4x+8.$

Step 2. Roots of the function.

Equate the function to zero.

$$\begin{gathered} f\left(x\right)=0 \\ {x}^3-2x^2-4x+8=0 \\ {x}^2(x-2)-4(x-2)=0 \\ (x^2-4)(x-2)=0 \\ x=2,-2. \end{gathered}$$

Roots of the function.

Step 3. critical points of the function.

Determine the first derivative of the function.

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}\left(x^3-2x^2-4x+8\right) \\ f\text{'}\left(x\right)=3x^2-4x-4 \end{gathered}$$

critical points of the function.

role="math" localid="1648666452201" $$\begin{gathered} f\text{'}\left(x\right)=0 \\ 3x^2-4x-4=0 \\ 3x^2-6x+2x-4=0 \\ 3x(x-2)+2(x-2)=0 \\ (3x+2)(x-2)=0 \\ x=2,\frac{-2}{3} \end{gathered}$$

f' is positive on interval $\left(-\infty ,\frac{-2}{3}\right)\&\left(2,\infty \right)$and negative on the interval$\left(\frac{-2}{3},2\right)$.

So f is increasing on intervals $\left(-\infty ,\frac{-2}{3}\right)\&\left(2,\infty \right)$and decreasing on the interval$\left(\frac{-2}{3},2\right).$

Step 4. concavity of the function.

find the second derivative of the function.

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=\frac{d}{dx}\left(3x^2-4x-4\right) \\ f\text{'}\text{'}\left(x\right)=6x-4 \end{gathered}$$

Roots of the second derivative of the function.

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=0 \\ 6x-4=0 \\ x=\frac{2}{3} \end{gathered}$$

f'' is positive on the interval $\left(\frac{2}{3},\infty \right)$and negative on the interval$\left(-\infty ,\frac{2}{3}\right).$

So f is concave up on interval $\left(\frac{2}{3},\infty \right)$and concave down on the interval $\left(-\infty ,\frac{2}{3}\right).$

So the sign chart is the following.

Step 5. Graph of function.

The graph of the function is the following.