f(x)= 0 if x is less than -5
f(x)= 2 if x is greater than or equal to -5 but less than 1
f(x)= -4 if x is greater than or equal to 1 but less than 4
f(x)= 0 if x is greater than or equal to 4
and g(x)= integrate from -5 to x of (f(t)dt)
Determine the value of each of the following:
a) The absolute maximum of g(x) occurs when x= ? and is the value ?
First, let's look at a graph of f:
When f is 0, then g is staying constant.
So g is constant when x<-5
When f is positive, then g is increasing. So g is increasing from x=-5 to x = 1.
When f is negative, then g is decreasing. So g is decreasing from x=1 to x=4.
And again, g is constant when x>4
Thus the maximum of g must occur at x=1.
`g(1)=int_(-5)^1 f(t) dt`
We can use calculus, but there is something easier:
An integral is just the area under the curve. Thus we want the area under f from x=-5 to x=1
f has a height of 2 there, and the width from -5 to 1 is 6. Thus the area is just the area of a rectangle: 2*6=12
So the maximum for g occurs at x=1, and there g is 12.