Video Transcript
If a function 𝑔 has its derivative 𝑔 prime of 𝑥 given as 𝑔 prime of 𝑥 equals three minus two 𝑒 to the power of negative 𝑥 cos of 𝑥, for 𝑥 is greater than negative three and less than three, on which of the following is 𝑔 increasing? Is it a) the open interval negative 0.957 to negative 0.592? b) The open interval from negative three to negative 0.592. c) The open interval from negative three to negative 0.957 and the open interval from negative 0.592 to three. Or d) the open interval from zero to three.
In this question, we’ve been given the function for the derivative of 𝑔 of 𝑥. That’s 𝑔 prime of 𝑥. And we’re being asked to establish on which of the open intervals the function 𝑔 itself is increasing. So we recall the definition for a function which is increasing.
A function is increasing when its derivative is greater than zero. So we need to work out at what points the function three minus two 𝑒 to the negative 𝑥 times cos of 𝑥 is greater than zero. And in fact, this isn’t a particularly nice inequality to solve. So instead, we’ll use our graphical calculators to plot the graph of 𝑦 equals three minus two 𝑒 to the power of negative 𝑥 cos of 𝑥. When we do, we obtain something that looks a little bit like this.
We said that the function is increasing when 𝑔 prime of 𝑥 is greater than zero. That’s these parts. And if we use our graphical calculator to find the point at which this graph crosses the 𝑥-axis, we see that that occurs when 𝑥 is equal to negative 0.957 and negative 0.592. So this means our function 𝑔 of 𝑥 is greater than zero when 𝑥 is greater than negative three and less than negative 0.957 and when 𝑥 is greater than negative 0.592 and less than three.
Comparing these to the values in our list, we see that the answer is c). It’s the open interval from negative three to negative 0.957 and the open interval from negative 0.592 to three.
