(a) State the Second Derivative Test for classifying a critical point of a…
MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
(a) State the Second Derivative Test for classifying a critical point of a twice–differentiable function.
(b) Explain why the test becomes inconclusive when the value of the second derivative at the critical point is zero.
(c) Provide one differentiable function f(x) and a real number c such that f′(c)=0 and f′′(c)=0. Apply the First Derivative Test to decide whether x=c is a point of local maximum, local minimum, or neither.
(d) Support your conclusion in part (c) by sketching the graph of f in a small neighbourhood of x=c, clearly indicating the behaviour of the slope on either side of the point.