Below is the graph of function f We define a new function by

Below is the graph of function f We define a new function by f(x) = integral_0^x f(x) dt. Answer the following questions about F and justify your answer. a) Is F(b) positive on negative? b) At which point or points does F have a local minimum c) Where is function F concave up? d) on the domain [-1, 5], at which point (points) does F have an inflection point? e) On the domain [-1, 5] at which point (points) does F have an global minimum? f) Find two values of x such that F(x) = 0.

Solution

a.

We can see on the graph when the horizontal axis reads 6 the graph is above the axis which means f(6) is positive

b.

Local minimums are points so that in a small neigbourhood of that point that point takes the smallest value. Though if one looks at a larger interval one many find other minimums also. So this is not a global minimum.

We can see on the graph there are 2 such points when x=2 and x=4

c.

Concave can be crudely considered as like U shape. So concave up is U shaped curve and concave down is U upside down.

We see from x=1 to x=2.5 the graph is concave up because at x=2.5 the graph starts to become concave down.

d.

Inflection point is a point where the graph changes from concave up to concave down.

It happens as we can see at x=1 and x=2.5 so these are the two inflection points

e.

To see this from the graph we look at the interval given and see where the graph has the smallest value which we can see is at x=2

f.

F(x)=0 happens when the graph meets x axis

We can see it meets x axis at x=3,5