Let be a twice–differentiable function whose first derivative satisfies at and
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let q be a twice–differentiable function whose first derivative satisfies q′(x)=0 at x=−1 and x=4. The second derivative of q is given analytically by q′′(x)=(x+1)(x−3)(x−6).
(a) For each critical point x=−1 and x=4, evaluate q′′ and determine whether the critical point is a local maximum, a local minimum, or neither. Provide justification.
(b) On which open intervals is the graph of q concave upward? Explain your reasoning.
(c) The point x=3 is a zero of q′′ but not of q′. Explain the graphical significance of x=3 and why it is not classified as a local extremum of q. Include a discussion of concavity.