00:01
In this problem, we are given a piecewise function y that is defined as 5x plus 3 for x less or equal than minus 1, or x squared minus 3 for x greater than minus 1.
00:13
Let's start by graphing our function.
00:16
What does this look like? so when x is less than minus 1, y looks like a linear curve.
00:31
When x is equal to minus 1, we have that the line is equal to minus 2.
00:42
And because here we have a line, we'll be increasing to the point x equal to minus 2, like so.
00:56
For x greater than minus 1, our curve is equal to y squared, x squared minus 3, which is a parabola.
01:08
So it looks something like this.
01:09
So this is our piecewise function, and we notice that we have a cusp at the coordinates minus 2 and minus 1.
01:21
So let's see if we have a critical point in the curve defined piecewise before or after the cusp.
01:32
We find critical numbers by calculating the first derivative of our function.
01:38
So y prime will be equal to 5 for x less or equal than minus 1, or will be equal to 2x for x greater than minus 1.
01:56
And we see that y prime is equal to 0 if and only if x is equal to 0 in our second function here, which means that here we have a critical point at x equal to 0, which is coming from a parabola which peaks at this point, or has a minima at this point.
02:23
I should have perhaps drawn a parabola something more like this, with a peak right here...