Let g(x)= (3x) (x) for .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let g(x)=arctan⁡(3x)arcsin⁡(x)g(x)=\frac{\arctan(3x)}{\arcsin(x)} for 0<x<1100<x<\tfrac1{\sqrt{10}}. (a) Verify that gg is differentiable on the stated interval, citing every condition you check. (b) Derive g′(x)g'(x) and write your result in fully simplified form. (c) Solve the equation g′(x)=0g'(x)=0 on 0<x<1100<x<\tfrac1{\sqrt{10}}. Express each solution exactly. (d) For each critical point found in part (c), determine whether gg attains a relative maximum, a relative minimum, or neither. Support your classification analytically.

Related theory

Browse the problem library