Need some help!

Question

The figure above shows the graph of the continuous function f. The regions A, B, C, D, and E have areas 5, 2, 16, 5, and 6, respectively. For -7≤ x ≤ 9, the function g is defined by g(x) = -6 + integral [-3, x] f(t)dt.

a) Is there a value of x, for -3≤ x ≤ 2, such that g(x) = 0? Justify your answer.

b) Find the absolute minimum value of g and the absolute maximum value of g on the interval -7≤ x ≤ 9.

c)

i) Find the integral of (2x+16)dx.

ii) Find the value of the integral [-7, -5] f(2x+16)dx,

Answer 1

So since g(x) = -6 + integral[-3, x] and you want g(x) = 0 the integral has to equal 6. The problem says that the area of the region C which is the section of the graph the integral covers is 16 which is larger than 6. So the x they are asking about should be whatever value between -3 and 2 that will give you an area of 6. So yes there is a value of x that will give you g(x) = 0. Since we don't know what f(t) is we can't find what that value of x is but the question only asked if that value exists not what it is.