What are the points of inflection of #f(x)=x+cosx # on the interval #x in [0,2pi]#?

1 Answer
Oct 1, 2016

A point of inflection is a point in which the function switches from being concave to convex, or vice versa. Our tool to verify if a function is concave or convex is the second derivative, more precisely its sign: if #f''(x)>0# then #f# is concave in #x#, otherwise it's convex.

So, look for the convex/concave switch is the same as looking for the positive/negative switch for the second derivative.

So, first of all, let's compute it: since the derivative of a sum is the sum of the derivatives, we have

#f(x) = x + cos(x)#

#f'(x) = 1 - sin(x)#

#f''(x) = - cos(x)#

And since we need to refer to the #[0,2pi]# interval, the cosine is positive in #[0,pi/2]#, negative in #[pi/2,(3pi)/2]#, and again positive in #[(3pi)/2,2pi]#.

The second derivative is #-cos(x)#, so it will change negative areas with positive ones, but the switch points will still be #pi/2# and #(3pi)/2#