Chapter 5.3, Problem 107E

 Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

  a.    Suppose f is a positive decreasing function, for x > 0. Then the area function $A\left(x\right)={\displaystyle {\int }_0^{x}f\left(t\right)dt}$ is an increasing function of x.

  b.    Suppose f is a negative increasing function, for x > 0. Then the area function $A\left(x\right)={\displaystyle {\int }_0^{x}f\left(t\right)dt}$ is a decreasing function of x.

  c.    The functions p(x) = sin 3x and q(x) = 4 sin 3x are antiderivatives of the same function.

  d.    If A(x) = 3x² – x – 3 is an area function for f, then B(x) = 3x² – x is also an area function for f.