Answer to Question #258780 in Calculus for Lily

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

Domain of the function is not given, so let us take

$t\in (0,2)$

$y=sin3t$

To find the maximum and minima, we need to find the $y\prime$ and set this to zero.

$$y = sin 3t$$

differentiate with respect to $t$

$$y\prime = 3 cos 3t = 0$$ $$3t = \frac {\pi}{2}, \frac {3\pi}{2}$$ $$t = \frac {\pi}{6}; \frac {\pi}{2} = 0.52; 1.57$$

Now,

$$y\prime\prime = - 9 sin 3t$$

$y\prime\prime (0.52)=-9sin3(0.52)=-9sin(1.56)=-8.999 \\y\prime\prime (0.52) < 0\\ So, y \space has \space maximum \space at \space t = 0.52$

Again,

$y\prime\prime (1.57)=-9sin3(1.57)=-9sin(4.71)=8.999 \\y\prime\prime (1.57) > 0\\ So, y \space has \space minimum \space at \space t = 1.57$

To find the Inflection point set $y\prime\prime = 0$

$- 9 sin 3t = 0\\ sin 3t = 0 \space at \space 3t = \pi$ or 3.14

inflection point is at $t = 1.04$

Now plotting the graph of function to show the turning point , maxima ,minima

Turning point of function is t = 1.04 for $t\in (0,2)$ . Turning point of function is the point where concavity of function is changes. It is also called point of inflection.