Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: h(x) = sin x + cos x

Find the local maxima and local minima, if any, of the following functions. Find
also the local maximum and the local minimum values, as the case may be:

(iv) $f(x) = sin x - cos x$

Answers (1)

G Gautam harsolia

Given function is
$h(x) = \sin x - \cos x\\ h^{'}(x)= \cos x + \sin x\\ h^{'}(x)= 0\\ \cos x + \sin x = 0\\ \cos x = -\sin x\\ x = \frac{3\pi}{4} \ \ \ \ \ \ as \ x \ \epsilon \ \left ( 0,2\pi \right )$
Now, we use second derivative test
$h^{''}(x)= -\sin x + \cos x\\ h^{''}(\frac{3\pi}{4}) = -\sin \frac{3\pi}{4} + \cos \frac{3\pi}{4}\\ h^{''}(3\frac{\pi}{4}) = -(\frac{1}{\sqrt2})-\frac{1}{\sqrt2}\\ h^{''}(\frac{\pi}{4})=- \frac{2}{\sqrt2} = -\sqrt2 < 0$
Hence, $\frac{\pi}{4}$ is the point of maxima and maximum value is $h\left ( \frac{3\pi}{4} \right )$ which is $\sqrt2$