Question

1. Calculate the turning points of the function y=sin3t using differential calculus
2. Show which are maxima, minima or points of inflexion using the second derivative

Answer 1

1. $y=\sin(3t)$

Domain: $(-\infin, \infin)$

Find the first derivative

$$y'=(\sin(3t))'=3\cos(3t)$$

Find the critical number(s)

$$y'=0=>3\cos(3t)=0$$

$$3t=\dfrac{\pi}{2}+\pi n, n\in \Z$$

$$t=\dfrac{\pi}{6}+\dfrac{\pi n}{3}, n\in \Z$$

$$y(\dfrac{\pi}{6}+\dfrac{2\pi m}{3}, m\in \Z)=1$$

$$y(\dfrac{\pi}{2}+\dfrac{2\pi k}{3}, k\in \Z)=-1$$

The turning points are

$$\big(\dfrac{\pi}{6}+\dfrac{2\pi m}{3}, 1\big), \big(\dfrac{\pi}{2}+\dfrac{2\pi k}{3},-1\big), m,k\in \Z$$

2.

Find the second derivative

$$y''=(3\cos(3t))'=-9\sin(3t)$$

$$y''(\dfrac{\pi}{6}+\dfrac{2\pi m}{3})=-9<0$$

$$y''(\dfrac{\pi}{2}+\dfrac{2\pi m}{3})=9>0$$

The points $\big(\dfrac{\pi}{6}+\dfrac{2\pi m}{3}, 1\big), m\in \Z$ are maxima.

The points $\big(\dfrac{\pi}{2}+\dfrac{2\pi k}{3}, 1\big), k\in \Z$ are minima.

$$y''=0=>-9\sin(3t)=0$$

$$3t=\pi l, l\in \Z$$

$$t=\dfrac{\pi l}{3}, l\in \Z$$

$$y(\dfrac{\pi l}{3})=0, l\in \Z$$

The points $\big(\dfrac{\pi l}{3}, 0\big), l\in \Z$ are points of inflection.