Question

Let f be a twice-differentiable function for all real numbers x. Which of the following additional properties guarantees that f had a relative minimum at x=c?
(A) f’(c) = 0
(B) f’(c) = 0 and f”(c) < 0
(C) f’(c) = 0 and f”(c) > 0
(D) f’(x) > 0 for x < c and f’(x) < 0 for x > c

*the answer is not B*

Answer 1

Answer:

(C) f’(c) = 0 and f”(c) > 0

Step-by-step explanation:

A minimum occurs where the first derivative is 0 (the tangent line is horizontal), and the second derivative is positive (concave up).  The simplest example of this is a positive parabola, like y = x², which has a relative minimum at its vertex.