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I'm working through a textbook exercise and am asked to find the intervals in which $g(x) = 5(x+3)^2-2$ is increasing and decreasing.

The solution provided is decreasing on $(-\infty, -3)$ and then increasing on $(-3, \infty)$.

I cannot see how to arrive at this conclusion. These are exercises at the end of a chapter on the transformation of functions though the author may be including bits from previous chapters, however I searched through recent chapters and there was only a section on determining where a function is increasing or decreasing based on a graph, so visually.

In this case does the question expect me to derive the solution based on formula? Is that possible? Or am I perhaps expected to draw a graph for some arbitrary values and then try to determine the increasing and decreasing points?

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    $\begingroup$ Did you mean $5(x\color{red}-3)^2-2,$ or did you mean $(-\infty,\color{red}-3)$ and $(\color{red}-3,\infty)$? $\endgroup$ – J. W. Tanner Sep 24 at 14:24
  • $\begingroup$ Hi thanks for pointing that out, I have updated my question to match the textbook now. $\endgroup$ – Doug Fir Sep 24 at 14:32
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The $-2$ at the end is just translating the function down, it will not change if it is increasing or decreasing. Than you have a parabola, so you need to look at the vertex, which is at $-3$, since you have $(x-(-3))^2$. This is similar to $x^2$, just translated horizontally from $0$ to $-3$. You know that the parabola has values of $+\infty$ at $x=\pm\infty$, so it's decreasing from $-\infty$ to the vertex at $-3$, and increasing afterwards.

P.S. see @JWTanner 's comment

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Hint: $$g'(x)=10(x+3)$$ so you have to solve $$g'(x)\geq 0$$ and $$g'(x)<0$$So compute $$g(x_2)-g(x_1)$$ with h $$x_1<x_2$$

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  • $\begingroup$ the question tag was precalculus $\endgroup$ – J. W. Tanner Sep 24 at 14:25
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If you know (or see) that translations do not modify monotonicity, you could start from the function $$ Y=5X^2 $$ and put $X=x-3$ and $Y=y+2$. A translation in the $y$ variable does not alter monotonicity inteervals. A translation in the $x$ variable just translates accordingly the monotonicity intervals. Since $Y=5X^2$ is decreasin on $(-\infty,0)$ and increasing on $(0,+\infty)$, then $y=5(x-3)^2-2$ is decreasing on ... and increasing on ...

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