Question

Help please! Please please please!(:
With the problem below,
Determine the following:
g(2)=____
g(4)=____
absolute max of g(x) is when x is ____ and is the value ____.

Answer 1

Let:
\[f(x)=0 , x<4 \] \[f(x)=5 , -4\le x <1\] \[f(x)=-2 , 1\le x <3\] \[f(x)=0 , x \ge3\]
and
\[g(x)=\int\limits_{-4}^{x} f(t) dt\]

Answer 2

really would help to draw a picture, which is just a bunch of horizontal lines

Answer 3

the integral is the area of the rectangle
so
\[\int_{-4}^1f(x)dx=4\times 5=20\] which is the largest it can be, because the stuff after than it negative (below the \(x\) axis)

Answer 4

To find the extrema of \(g(x)\), apply the first derivative test.

If \(\displaystyle g(x)=\int_{-4}^xf(t)~dt\), then \(g'(x)=f(x)\).

Find your critical points (i.e. when \(f(x)=0\)), which shouldn't be too hard, then determine the intervals on which \(g\) is increasing/decreasing.

Answer 5

therefore the max is \(20\) when \(x=1\)

Answer 6

this max is not 20

Answer 7

hmm i think the critical points are not where the derivative is 0, but rather where the derivative is undefined
small matter though

Answer 8

oh of course, sorry, the max is \(5\times 5=25\) my mistake

Answer 9

@satellite73, both, actually: http://en.wikipedia.org/wiki/Critical_point_(mathematics) (first sentence)

Answer 10

@SithsAndGiggles i mean in this example

Answer 11

Right, I was thinking more generally :P

Answer 12

I'm so confused. so how is it determined at g(2) and g(4)?

Answer 13

in general of course critical point is where derivative is zero or undefined
in this example the important critical point would be at the change in the definition of the function, where the derivative would not be defined

Answer 14

take the area of the rectangle
at \[g(2)=\int_{-4}^2f(x)dx\] you have \(25\) for the area of the rectangle with base 5 and height 5, then \(-2\) for the area of the rectangle with base 1 and "height" \(-2\)

Answer 15

therefore \(g(2)=23\)

Answer 16

so with that g(4)=21 right?

Answer 17

sorry i got closed out

Answer 18

yea, \(g(4)=21\) right

Answer 19

yay! thanks!!(: