Problems from Increasing and decreasing functions

Development:

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: $$y^{\prime }=3x^2$$ 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 $y^{\prime }=0$.

For positive values of $x$, the derivative $y^{\prime }=-1$.

Therefore, $y^{\prime }\le 0$ in an environment to $x=0$, and therefore the function is decreasing at $x=0$ (it is not strictly decreasing!)

Solution:

a) Strictly increasing

b) Decreasing

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