Study the inceasing and decreasing intervals of the following functions:
a) $$y=x^2$$
b) $$y=\sin x$$
Development:
We will solve these two problems by using frist more graphical/intuitive tools, while also confirming our intuitons by studying the derivative.
a) See the graph of the function
Increasing intervals: $$$x\in (-\infty,0) \ \mbox{Strictly decreasing}$$$ $$$x\in(0,\infty) \ \mbox{Strictly increasing}$$$
If we compute the derivative: $$y'=2x$$
Therefore, for the points $$x < 0$$ the derivative is strictly negative, which implies that the function is strictly decreasing.
For the points $$x > 0$$ the derivative is strictly positive, or in this interval the function is strictly increasing.
b) See first the graph
Intuitively we can see that there are increasing and decreasing intervals that can be found periodically. We need the analytical tools to define with accuracy the above mentioned intervals.
If we compute the derivative: $$y'=cos(x)$$.
Intervals with $$y' > 0: \ \Big(-\dfrac{\pi}{2},\dfrac{\pi}{2}\Big),\Big(\dfrac{3\pi}{2},\dfrac{5\pi}{2}\Big),\Big(\dfrac{7\pi}{2},\dfrac{9\pi}{2}\Big),\ldots$$
Intervals with $$y' < 0: \ \Big(\dfrac{\pi}{2},\dfrac{3\pi}{2}\Big),\Big(\dfrac{5\pi}{2},\dfrac{7\pi}{2}\Big),\Big(\dfrac{9\pi}{2},\dfrac{11\pi}{2}\Big),\ldots$$
In fact there are infinite increasing and decreasing intervals.
Solution:
a) $$x\in (-\infty,0): \ \mbox{strictly decreasing}$$; $$x\in(0,\infty): \ \mbox{strictly increasing}$$
b) Strictly increasing $$y' > 0: \ \Big(-\dfrac{\pi}{2},\dfrac{\pi}{2}\Big),\Big(\dfrac{3\pi}{2},\dfrac{5\pi}{2}\Big),\Big(\dfrac{7\pi}{2},\dfrac{9\pi}{2}\Big),\ldots$$
Strictly decreasing $$y' < 0: \ \Big(\dfrac{\pi}{2},\dfrac{3\pi}{2}\Big),\Big(\dfrac{5\pi}{2},\dfrac{7\pi}{2}\Big),\Big(\dfrac{9\pi}{2},\dfrac{11\pi}{2}\Big),\ldots$$