A differentiable function is defined for all real numbers and satisfies g'(x)= 6x-4…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

A differentiable function

g

is defined for all real numbers and satisfies

g'(x)=\frac{6x-4}{x^{2}-4x+13},\quad g(0)=5.

Use this information to analyze the behavior of

g

. (1) Find an explicit expression for

g(x)

. Show all work, including the constant of integration determined by the initial condition. (2) Determine all critical points of

g

and classify each as a relative maximum, relative minimum, or neither. Support your answer with appropriate reasoning. (3) Compute

\displaystyle\int_{0}^{3}g'(x)\,dx

and state its numerical value. Then use the result to find

g(3)

. Explain how your answer is consistent with the net change theorem.