Question

You have been asked to determine whether the function f(x)= 3 + 4cosx +cos2x is ever negative.

a) Explain why you need to consider values of x only in the interval [0,2π ].
b) Is f ever negative? Explain.

This is extra credit and I can't seem to figure it out. I'm used to find the derivative of polynomials.

Answer 1

I hope you have come across the identity
cos2x = 2 cos^2 x - 1
so
3 + 4cosx +cos2x
= 3 + 4cosx + 2cos^2x - 1
= 2cos^2x + 4cosx + 2
= 2(cos^2x + 2cosx + 1)
= 2(cosx+1)^2

something squared >0
so 2(something squared)>0
and f(x) can thus never be negative

the above answers part b)

for a) the period of cosx is 2π and the period of cos2x is π

so the period of the whole function is 2π
So whatever happens will happen between 0 and 2π, and simply repeat itself after that.

Answer 2

what do you mean by something squared?