Chapter 3.6, Problem 34E

To determine

To find: the critical point of $y$ and explain the transformation of each new critical point.

Answer to Problem 34E

  $x=-3$

The given critical point is relative maximum of f, the second critical point is relative minimum of f and third critical point is relative minimum of y.

Explanation of Solution

Given information:

  $\begin{array}{l}y=2f\left(x+5\right)-1\\ x=2\left[\text{Relative maximum of }f\right]\\ x=-1\left[\text{Point of inflection of }f\right]\end{array}$

Calculation:

Since, the function is multiplied by $-2$ , essentially it flips it over and doubles its distance. So, the relative maximum point would become relative minimum and it would be twice its distance (stretch) or $y$ value. The graph will look as follows:

  

Then the part $-2f\left(x+5\right)-1$ moves 5 units left and 1 unit down. The final graph will be as follows:

  

The critical point of $y$ is at $x=-3$ i.e. relative minimum of y.