Review AP Calculus AB 5.7 by evaluating the second derivative at critical points and using its sign to classify local maxima or minima.
Given the following differential equation tart fraction numerator d y over denominator d x end fraction equals x minus 2 y plus 1
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.