The number of disjoint intervals over which the function $f(x) = |0.5x^ 2−| x | |$ is decreasing is
- A)one
- B)two
- C)three
- D) none of these
I actually solved it and the answer is three but I want to know if there is an easier method to solve this.
My approach was to break down the absolute function into piecewise function.
I first did this:
$$ f(x) = \left\{ \begin{array}{ll} 0.5x^2-|x| \quad |x| > 2 \\ -0.5x^2+|x| \quad |x|<2 \end{array} \right. $$
then I break it even further for $|x|$:
$$ f(x) = \left\{ \begin{array}{ll} 0.5x^2+x \quad x < -2 \\ -0.5x^2-x \quad -2<x<0 \\ -0.5x^2+x \quad 0<x<2 \\ 0.5x^2-x \quad x>2 \end{array} \right. $$
Then I differentiated each of these and put the values to find if $f'(x)$ is positive or negative in this interval and found that the function is decreasing in three intervals.
But this process is very long, I want to know a more straightforward way also if I use a graph how would I go about constructing it?
Hint: for $x\ne 0$ we have $|x|'=\frac{x}{|x|}$. You can use this and the chain rule to get an explicit formula for the derivative (where it is differentiable).