Question

Find the local max and min values of f.
\[f(x)=12-2\left| x+3 \right|\]

Answer 1

we have to explicit the absolute value

Answer 2

hint:
\[\begin{gathered}
\left| {x + 3} \right| = x + 3,\quad x \geqslant - 3 \hfill \\
\hfill \\
\left| {x + 3} \right| = - \left( {x + 3} \right),\quad x < - 3 \hfill \\
\end{gathered} \]

Answer 3

How do I know which one to use? @Michele_Laino

Answer 4

both methods are good.
nevertheless if you want to rewrite the equation of your function, you have to substitute the absolute value, as I have indicated above

Answer 5

for example, if
\[x \geqslant - 3\]
then your function can be rewritten as below:
\[12 - 2\left( {x + 3} \right) = 12 - 2x - 6 = 6 - 2x\]
which is represented by the right branch of the drawing of @Jhannybean
whereas, if:
\[x < - 3\]
then your function can be rewritten as below:
\[12 - 2\left[ { - \left( {x + 3} \right)} \right] = 12 + 2x + 6 = 18 + 2x\]
which is represented by the left branch of the drawing of @Jhannybean

Answer 6

oh okay, so then I take the derivatative of one of them? @Michele_Laino

Answer 7

mmm

look at the graph

do you need to do anything else to find max and mins?

Answer 8

It is suffice to keep in mind the graph of @Jhannybean

Answer 9

what is the maximum value?

Answer 10

when f(-3)=12 @Michele_Laino

Answer 11

from the graph...
But would you get the -3 from?

Answer 12

algebraically?

Answer 13

ok! That's right!