Answer the following questions related to local extrema of real-valued functions.
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Answer the following questions related to local extrema of real-valued functions.
(i) Define a critical point of a differentiable function.
(ii) State the conditions of the second derivative test for identifying a local maximum or a local minimum at a critical point.
(iii) Explain why the second derivative test may fail at some critical points and describe the additional step that must be taken when the test is inconclusive.
(iv) For the function f(x)=x3, locate the critical point and classify its nature by applying your answers in parts (ii) and (iii).
(v) Summarise, in logical order, the complete procedure for finding and classifying all local maxima and minima of any polynomial defined on a closed interval.