Suppose that f has a continuous second derivative for all x, and that f(0)=1, f'(0)=2, and f''(0)=0.

Question

A. Does f have an inflection point at x=0? Explain your answer.

B. Let g'(x) = (3x^2 + 2)f(x) + (x^3 + 2x + 5)f'(x). The point (0,5) is on the graph of g. Write the equation of the tangent line to g at this point.

C. Use your tangent line to approximate g(0.3).

D. Find g''(0).

Answer 1

A yes, if f"=0 and f'≠0
B find g'(0)
Then the tangent line is
y-5 = g'(0) (x-0)

Answer 2

idk