If ℎ prime of 𝑥 equals three 𝑥 minus two 𝑥 squared plus the square root of four 𝑥 squared plus four, what is the value of 𝑥 at the relative maximum of ℎ of 𝑥?
Let’s begin by recalling what we actually know about the relative extrema for functions. A function achieves a relative maximum at points at which the graph changes from increasing to decreasing. Similarly, to choose a relative minimum at the points where the graph changes from decreasing to increasing. And so a relative maximum might look a little something like this.
Before the relative maximum, the function is increasing. In other words, the derivative of that function is greater than zero. At the relative maximum, the derivative is equal to zero. And we can see, after the relative maximum, the derivative of our function, 𝑓 prime of 𝑥, is less than zero. So to find the relative maximum of ℎ of 𝑥, we’re going to draw the graph of its derivative, ℎ prime of 𝑥.
We’re then going to look for the point on our graph of ℎ prime of 𝑥 where it changes from being positive to negative. And we’ll use our graphical calculators to achieve this. When we plot the graph of 𝑦 equals three 𝑥 minus two 𝑥 squared plus the square root of four 𝑥 squared plus four, we end up with something that looks a little like this. And we see that the graph changes from being positive to negative here. We use our calculator to find the coordinate of this root. And we find it to be equal to 2.5730. And so this means the value of 𝑥 at the relative maximum of ℎ of 𝑥 is 2.573.