maximum/minimum question..

[ihatecalc]

a) If I am given the graph of g', how do I find its local maxima and local minima?

b) Also, the question states: the graph of g' has a local maximum at x=3.8 and a local minimum at x = 7.4. What do these facts say about the graph of g?

please help me im stuck!!! especially part A!!!

 

[skeeter]

(a) max for g(x) will be located where g'(x) changes sign from positive to negative.
min for g(x) will be located where g'(x) changes sign from negative to positive.
now ... can you tell me why?

(b) if g'(x) has a local max or a local min, then g"(x) = 0 and g"(x) is changing signs ... what kind of point will you have on the graph of g(x) when those conditions exist?

 

[soroban]

Hello, ihatecalc!

A sketch should help . . . Then baby-talk your way through it.

a) If I am given the graph of \(\displaystyle g'\), how do I find its local maxima and local minima?

Suppose the graph of \(\displaystyle g'(x)\) looks like this:

            |
            | *
            |
            |  *
            |   *
            |     *
      - - - + - - - -*- - - - - - -
            |        P    *
            |                   *
            |

Note: This is not the graph of \(\displaystyle g(x)\), the original function.
. . . . .This gives us the behavior of the slope of \(\displaystyle g(x)\).


The \(\displaystyle x\)-intercepts are the local maxima and minima.
. . Think about it . . . That's where \(\displaystyle g'(x)\,=\,0\) ... horizontal tangents.

To the left of \(\displaystyle P\), the graph is positive.
. . Since \(\displaystyle g'(x)\) is positive, the slope is positive: \(\displaystyle \nearrow\)

To the right of \(\displaystyle P\), the graph is negative.
. . Since \(\displaystyle g'(x)\) is negative, the slope is negative: \(\displaystyle \searrow\)


So at \(\displaystyle P\), the graph is shaped like this: \(\displaystyle \nearrow\;^{\longrightarrow}\;\searrow\)

. . Therefore, there is a local maximum at \(\displaystyle P.\)

(If takes a bit of practice, but it's not difficult.)