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- Derivatives
- Increasing and decreasing functions
- Ejercicios
Increasing and decreasing functions
Study if the following functions are increasing / decreasing at point $x=0$.
a) $y=x^3$
b) $y= \left\{ \begin{array} {rcl} 0 & \mbox{ if } & x \leq 0 \\ -x & \mbox{ if } & x>0 \end{array}\right.$
a) See the graph
At a first glance we see that the graph is an increasing function, although in $x=0$ it is less clear what is happening. Is it, then, a strictly increasing function in $x=0$?
Let's calculate it analytically. For that let's calculate the derivative: $$y'=3x^2$$ Let's see what the sign of the derivative is in the points placed in a environment to $x=0$.
We can see that for any value of $x$ (different from zero) the derivative is positive. Therefore all the points of the environment of $x=0$ have positive derivative. This means that the function is strictly increasing in $x=0$.
b) See the derivative in values close to $x=0$.
For negative values of $x$, the derivative $y'=0$.
For positive values of $x$, the derivative $y'=-1$.
Therefore, $y'\leq0$ in an environment to $x=0$, and therefore the function is decreasing at $x=0$ (it is not strictly decreasing!)
a) Strictly increasing
b) Decreasing