Chapter 5.5, Problem 10QR

To determine

To check: whether the given curve is concave up or concave down on the given interval.

Answer to Problem 10QR

Concave down.

Explanation of Solution

Given:

  $y=\sin x-\cos x$ on $\left[1,2\right]$ .

Concept used:

A curve $y=f\left(x\right)$ is said to be concave down on the interval $\left[a,b\right]$ if $f^{″}\left(x\right)<0$ on $\left[a,b\right]$ and concave up if $f^{″}\left(x\right)>0$ on $\left[a,b\right]$ .

Calculation:

Consider the given equation,

  $y=\sin x-\cos x$

Differentiating with respect to $x$ , it follows that

  $\begin{array}{c}\frac{dy}{dx}=\cos x-\left(-\sin x\right)\\ =\cos x+\sin x\end{array}$

Again differentiating with respect to $x$ , it follows that

  $\frac{d^{2}y}{d{x}^2}=-\sin x+\cos x$

Now,

  $\begin{array}{c}\sin x>0\text{on}\left[1,2\right]\text{and}\cos x>0\text{on}\left[1,2\right]\\ -\sin x<0\text{on}\left[1,2\right]\text{and}\cos x>0\text{on}\left[1,2\right]\\ -\sin x+\cos x<0\text{on}\left[1,2\right]\end{array}$

So, $\frac{d^{2}y}{d{x}^2}<0$ on $\left[1,2\right]$ .

Hence the curve is concave down on $\left[1,2\right]$ .