Question

The graph of the function f is shown above. Let g be the function defined by g left parenthesis x right parenthesis equals integral subscript 1 superscript x f left parenthesis t right parenthesis d t . At what values of x in the interval 0.5 < x < 6.5 does g have a relative maximum?

Answer 1

The function g(x), defined as the integral of f(t), has a relative maximum where f(x) has a peak. Given the graph, this occurs at x = 3 and x = 6, resulting in answer (C) 2 and 5.

The graph where the function f has local minima at x = 1 and x = 4, and local maxima at x = 3 and x = 6.

Now, you want to find the values in the interval 0.5 < x < 6.5 for which the function
`g(x)= \int\limits^x_0 {f(t)} \, dt` has a relative maximum.

The relative maximum for g(x) would occur where the derivative of g(x) is equal to zero. In this case, since g(x) is the integral of f(t), the relative maximum of g(x) would occur where f(x) has a relative maximum.

From your description, f(x) has a relative maximum at x = 3 and x = 6. Therefore, the correct answer is: (C) 2 and 5

The complete question is:

The graph of the function f is shown above. Let g be the function defined by g(x) = of z in the interval 0.5<x< 6.5 does g have a relative maximum? f(t)dt. At what values

(A) 3 only

(B) 4 only

(C) 2 and 5

(D) 1, 3, 4, and 6