Consider the function: 𝑓(𝑥) = −3𝑥^3 +(𝑎 + 𝑏)𝑥 + 1

Question

a. State the derivative

b. Find all points on the graph of the function that have a slope of 𝑏. (Hint: You need to solve an equation.)

Answer 1

The derivative equals f'(x)=-9x^2+a+b. In order for the slope to equal b we need to have that -9x^2+a+b=b which is the same as 9x^2=a. Hence, a>=0 in order for the last equation to make sense. Then x equals x=sqrt(a)/3 or x=-sqrt(a)/3.