Calculus problem?

Question

Given the function f defined for all real numbers x by f(x) = 2 |x - 1|x^2.
a) What is the range of the function?
b) For what values of x is the function continuous?
c) For what values of x is the derivative of f(x) continuous.
d) Determine the value of I = integral (1; 0) f(x)dx.

Answer 1

The range is obviously [0,infinity)

|a continuous function| is always continuous.

|a differentiable function| is always differentiable except possibly (usually but not always) at its zeros.

When doing calculus with absolute values, just write something as two separate functions -- in this case 2(x-1)x^2 and -2(x-1)x^2. Then check on which intervals it is which function, and do the calculus separately on each of those intervals.

There. That's enough to get you quickly to your answers.