How do you find all points of inflection given #y=-2sinx#?

Question

Paisley Johnson · Accepted Answer

There is a point of inflection whenever #-2sinx=0#

Points of inflection occur when the curve changes concavity. Since this is a sine wave, there are an infinite number of points of inflection.

A function is concave up when the second derivative (#f''#) is greater than 0, and concave down when the second derivative is below 0. Critical points, therefore, are when the second derivative equals 0.

Differentiate #y=-2sinx# to get #y'=-2cosx#. Differentiate again to get #y''=2sinx#, the original function.

Whenever #-2sinx=0#, there is a point of inflection. This can be intuitively verified by graphing #y=-2sinx#.

graph{-2sinx [-pi, pi, -3, 3]}

Whenever the curve crosses the x-axis (that is, whenever y=0), the concavity changes.

Christopher Agresta · Answer

undefined To find all points of inflection for the function $ y = -2 \sin(x) $, you need to locate where the second derivative changes sign.

First, find the first derivative of $ y $ with respect to $ x $, then find the second derivative. Set the second derivative equal to zero and solve for $ x $ to find critical points. Next, determine the intervals where the second derivative changes sign by testing points in each interval.

The points where the second derivative changes sign correspond to points of inflection for the function.