Find Max value ..

Question

Given , $\ P = \cos^4 \alpha + \sin^4 \alpha + 1$ and $\ Q = \cos^6 \beta + \sin^6 \beta + 1$ , where $\alpha$,$\beta \in \Bbb R$ and are independent.

How to find the value of $\ (P+Q)_{max}$ ?

Answer 1

First, I assume that $\alpha,\beta$ in your problem are independent, so $$(P+Q)_{max}=P_{max}+Q_{max}$$ Let $\cos^2x=k$. Hence, $$P=k^2+(1-k)^2+1$$ and $$Q=k^3+(1-k)^3+1$$ Since we know $k\in[0,1]$ we can differentiate $P, Q$ w.r.t $k$ to find the maximum value.

It turns out that $$(P+Q)_{max}=P_{max}+Q_{max}=(1+1)+(1+1)=4$$

Answer 2

Without derivatives:

We have

$$c^4+s^4=(c^2+s^2)^2 -2c^2s^2=1-2c^2s^2$$

and

$$c^6+s^6=(c^2+s^2)(c^4-c^2s^2+s^4)=(c^2+s^2)^2-3c^2s^2=1-3c^2s^2.$$

This tells us that the maximum of the sum is $(1-2\cdot0+1)+(1-3\cdot0+1)=4$.