Writing Assignment
In this project we are given with a graph of continuous function f which is defined on [0, 10]. Also, given that:
We have to answer on the number of questions regarding function g properties based on the information given about function f.
We begin with determination what is g(0)? Since g(x) is:
gx=0xftdt
Hence
g0=00ftdt≡0
It’s zero, because the integral’s limits are equal.
Consider g10:
g10=010ftdt
The function value at this point is an integral from f(x) among its domain – from 0 to 10. As we know, the geometric interpretation of the integral is the area of under the graph of the function.
That’s why if we consider the areas under the graph above x-axis and below x-axis, we can see that the total area under the graph above the x-axis (3 intervals – from 0 to 2, from 4 to 6 and from 8 to 10) is bigger than the area below it (2 intervals – from 2 to 4 and from 6 to 8). The value of integral is sum of the areas above minus sum of areas below. Hence, g(10) will be positive.
It is known that:
g'x=0xftdt'=fx
That’s why g’(x) is 0 in all points where f=0. The points are x=2, 4, 6, 8, 10
And these points are extremum points. x=2, 6, 10are local maximums, x=4,8 are local minimums.
The second derivative g’’(x) is:
g''x=f'x
Where g’’(x) is negative, there a graph is concave up, where positive – it is concave down.
The sings of f’(x) are positive on (0, 1), (3, 5) and (7, 9) and negative on (1, 3), (5, 7) and (9, 10).
Now we are able to draw a possible graph taking in consideration the analysis mentioned above.
