Question

How do you determine the interval(s) over which `f(x)` is increasing, decreasing, constant, concave up, concave down? What are the zeroes of `f(x)`? For what values of `f(x)` discontinuous?

Assume `f(x)` is a standard parabola of `y=x^2`.

Answer 1

See below

Explanation:

To determine where the function is increasing, decreasing or constant you have to look at the derivative, more specifically at its sign. If the derivative is positive the function is increasing, if the derivative is negative the function is decreasing, and if the derivative is zero on a whole interval, the function is constant (note that if the derivative is zero for a single point then it's just a point of maximum/minimum, so you can't say that the function is constant).

The derivative of `x^2` is `2x`, and the sign of `2x` is the same as that of `x`. So, the function is decreasing for `x<0`, increasing for `x>0`, and has a minimum at `x=0`.

As for the concavity, you have to do something similar with the second derivative: if positive, the function is concave up, if negative concave down, if zero it's switching between the two.

The second derivative is the derivative of the derivative, and the derivative of `2x` is `2`, which is (always) positive. So, the parabola is always concave up.

To find the zeroes, simply set the equation

`f(x)=0 \iff x^2=0 \iff x = \pm\sqrt(0) = 0`

So, the only (double) zero of `f` is `x=0`

Finally, `f(x)` is a polynomial, and as such it is always continuous.