Use your result from part b to determine all of the points

Question

1. Let f{x) = ¼ x³ + 4.

a. Sketch a graph of y = f(x) and explain why f is an invertible function.

b. Let g be the inverse of f and determine a formula for g.

c. Compute f'(x), g'(x), f'(2), and f'(6). What is the special relationship between f'(2) and g'(6)? Why?

2. Consider the curve defined by the equation x = y⁵ - 5y³ + 4y, whose graph is pictured in Figure.

a. Explain why it is not possible to express y as an explicit function of x.

b. Use implicit differentiation to find a formula for dy/dx.

c. Use your result from part (b) to find an equation of the line tangent to the graph of x = y⁵ - 5y³ + 4y at the point (0,1).

d. Use your result from part (b) to determine all of the points at which the graph of x = y⁵ - 5y³ + 4y has a vertical tangent line.

Figure: The curve x = y⁵ - 5y³ + 4y.