We can determine where a function is convex or concave, by using the second derivative. If:
(d^2y)/(dx^2)f(x)>0color(white)(888) Convex( concave up )
(d^2y)/(dx^2)f(x)<0color(white)(888) Concave( concave down )
The second derivative is just the derivative of the first derivative. .i.e.
(d^2y)/(dx^2)f(x)=dy/dx(dy/dxf(x))
For f(x)=cos(x)
dy/dx=-sin(x)
(d^2y)/(dx^2)(cos(x))=dy/dx(-sin(x))=-cos(x)
Now we solve the inequalities:
-cos(x)>0color(white)(888)[1]
-cos(x)<0color(white)(888)[2]
-cos(x)>0color(white)(888)[1]
pi/2 < x< (3pi)/2
-cos(x)<0color(white)(888)[2]
0 < x < pi/2color(white)(88) , (3pi)/2 < x < 2pi
Notice that bb((3pi)/2) is on the point where the function changes from convex to concave. This is called a point of inflection ( inflexion in the UK ), so at bb((3pi)/2) it is neither concave nor convex.
This is verified by its graph:
![enter image source here]()
