AP Calculus AB 5.7 Second Derivative Test Question Bank
Practise second-derivative-test questions by evaluating curvature at critical points and classifying local maxima or minima.
- Syllabus
- Effective Fall 2025
- Course
- AP Calculus AB
Exam points
- evaluate the second derivative at a critical point
- use concavity to classify a local maximum or minimum
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.7 question 1
Given the following differential equation
tart fraction numerator d y over denominator d x end fraction equals x minus
2 y plus 1
(c) The function y=f(x) is the solution to the differential equation with
initial condition f(-1)=0. Determine whether f has a local maximum,
local minimum, or neither at x=-1. Justify your answer.
(c) (-1,0) is a critical point because
art fraction numerator d y over denominator d x vertical line
from left parenthesis negative 1, 0 right parenthesis equals
negative 1 minus 2 left parenthesis 0 right parenthesis plus 1
equals 0
. And
art fraction numerator d squared end exponent y over
denominator d x squared end exponent vertical line from left
parenthesis negative 1, 0 right parenthesis equals 4 left
parenthesis 0 right parenthesis minus 2 left parenthesis
negative 1 right parenthesis minus 1 equals 1 greater than 0
, so by the Second Derivative Test, the solution curve f(x) has a
local minimum.
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