Chapter 6.5, Problem 2QR

To determine

To verify the concavity of the given function on the given interval.

Answer to Problem 2QR

Concave up.

Explanation of Solution

Given:

  $y=x^4-12x-5,\left[8,17\right]$

Calculation:

  $y=x^4-12x-5$

Differentiating with respect to x, we get

  $\begin{array}{l}{y}^{\prime }=4x^3-12\left(1\right)-0\\ \Rightarrow y^{\prime }=4x^3-12\end{array}$

Again differentiating with respect to x, we get

  $\begin{array}{l}{y}^{″}=4\left(3x^2\right)-0\\ \Rightarrow y^{″}=12x^2\end{array}$

Now,

  $\begin{array}{l}{x}^2>0\text{for}\text{all}x\in R.\\ \Rightarrow 12x^2>0\text{for}\text{all}x\in R.\\ \Rightarrow y^{″}\text{is}\text{positive}\text{for}\text{all}x\in R.\\ \Rightarrow y^{″}\text{is}\text{positive}\text{for}\text{all}x\in \left[8,17\right].\end{array}$

Hence the curve $y=x^4-12x-5$ is concave up on the interval $\left[8,17\right]$ .