Answers:
- Smaller x value = -3
- Larger x value = -1
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Explanation:
Apply the derivative (see attached image for the steps)
You should get [tex]\frac{dy}{dx} = -\frac{1}{(x+2)^{2}}\\\\[/tex]
The line y = x is really y = 1x+0. We see that it's in the form y = mx+b with m = 1 as the slope. The negative reciprocal of this is m = -1, which is the perpendicular slope. As a rule, any two perpendicular lines always have their slopes multiply to -1 as long as neither line is vertical or horizontal.
So we want the derivative we just found to be set equal to -1, that way we have the x value lead to this derivative slope value.
We want dy/dx = -1
In short, the equation we need to solve is [tex]-\frac{1}{(x+2)^{2}} = -1\\\\[/tex]
Let's solve for x
[tex]-\frac{1}{(x+2)^{2}} = -1\\\\\frac{1}{(x+2)^{2}} = 1\\\\(x+2)^{2} = 1\\\\x+2 = \pm\sqrt{1}\\\\x+2 = \pm1\\\\x+2 = 1 \ \text{ or } \ x+2 = -1\\\\x = -2+1 \ \text{ or } \ x = -2-1\\\\x = -1 \ \text{ or } \ x = -3\\\\[/tex]
The smaller x value is -3 while the larger x value is x = -1.
