Question

Let g be the function given by g(x) = the integral from 0 to x sin(t^2) for -1 < or equal to x < or equal to 3. Find the interval where the function g(x) is decreasing.

Answer 1

To find the interval where the function g(x) is decreasing, we need to find the derivative of g(x) and determine where it is negative.

By the Fundamental Theorem of Calculus, we have:

g'(x) = sin(x^2)

Since sin(x^2) is an oscillating function, we need to examine its behavior on different intervals.

For x in the interval [-1, 1], sin(x^2) is positive and decreasing.

For x in the interval [1, sqrt(pi/2)], sin(x^2) is negative and decreasing.

For x in the interval [sqrt(pi/2), sqrt(2)], sin(x^2) is negative and increasing.

For x in the interval [sqrt(2), sqrt(3)], sin(x^2) is positive and increasing.

Therefore, the interval where g(x) is decreasing is [1, sqrt(pi/2)].