Question

Use a sign chart for $f\text{'}\text{'}$to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility

$f\left(x\right)=(x-2{)}^4$

Short answer

The function has no inflection points and is concave up on both$(-\infty ,2)\text{and}(2,\infty ).$

Step-by-step solution

Step 1. Given Information.

The given function is$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{4}}$.

Step 2. Second derivative.

On differentiating the given function, we get,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\frac{d}{dx}(x-2{)}^4 \\ =4(x-2{)}^{4-1}\frac{d}{dx}(x-2) \\ =4(x-2{)}^3 \\ {f}^{\text{'}\text{'}}(x)=\frac{d}{dx}4(x-2{)}^3 \\ =12(x-2{)}^2 \end{gathered}$$

Step 3.Sign Chart.

Now,

$f^{\text{'}\text{'}}\left(x\right)=0\text{when}x=2$,

Therefore, the sign chart will be,

Since, the sign does not change there is no inflection point.

And the function is concave up in$(-\infty ,2)\text{and}(2,\infty ).$

Step 4.Verification.

The graph of the function is ,