Determining Intervals on Which a Function Is Increasing or Decreasing Quiz

Let f be the function defined by f(x)=(sin⁡x)e^−x on the interval [−pi/2,pi/2]. On which of the following open intervals is f increasing?

(-pi/2,pi/4)

Let f be the function defined by f(x)=(x+x^2)e^-2x. On which of the following open intervals is f increasing?

(-√2/2,√2/2)

Let f be the function with the derivative given by f'(x)=sinx+cos(2x)-pi/4 for 0≤x≤pi. On which of the following intervals is f increasing?

[0,0.724] and [2.418,3.142]

Let f be the function with derivative given by f'(x)=sinx+xcosx for 0≤x≤pi. On which of the following intervals is f increasing?

[0,2.029]

Let f be the function with derivative given by f'(x)=x^2-(a+b)x+ab=(x-a)(x-b), where a and b are constants such that a<b. Which of the following statements is true?

f is decreasing for a<x<b because f'(x)<0 for a<x<b.

Let f be the function with derivative given by f′(x)=x^2−a^2=(x−a)(x+a), where a is a positive constant. Which of the following statements is true?

f is decreasing for −a<x<a because f′(x)<0 for −a<x<a.