Chapter 6.5, Problem 1QR

To determine

To find: that the curve is concave up or concave down on the given interval.

Answer to Problem 1QR

Concave down

Explanation of Solution

Given:

  $y=\text{cos }x$

Calculation:

  $\begin{array}{c}y=\text{cos }x\\ y\text{'}=-\sin x\\ y"=-\cos x\end{array}$

  $\cos x>0\text{for}\text{all}x\text{in }\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\text{so}\cos x>0\text{for}\left[-1,1\right].$

Therefore $y"=-\cos x<0\text{for}\text{all}x\in \left[-1,1\right],$ the curve is concave down.