find-minima-of-x-1-x-2-2-5-1-x-1-2-4x-2-x-1-x-2-R-

Question Number 70167 by Kunal12588 last updated on 01/Oct/19

f i n d m i n i m a o f

{(x_{1} - x_{2})}^{2} + 5 + \sqrt{1 - {(x_{1})}^{2}} + \sqrt{4 x_{2}} \forall x_{1}, x_{2} \in R

Commented by MJS last updated on 02/Oct/19

- 1 ⩽ x_{1} ⩽ 1 \Rightarrow x_{1} = \sin p

0 ⩽ x_{2} \Rightarrow x_{2} = q^{2}

f (p, q) = {(q^{2} - \sin p)}^{2} + 5 + ∣ \cos p ∣ + 2 q

{it}^{'} s easy to see no maximum exists :

q \to + \infty \Rightarrow f (p, q) \to + \infty

searching the minimum :

0 ⩽ {(q^{2} - \sin p)}^{2}

the minimum is at q^{2} = \sin p \Leftrightarrow p = 2 n π \pm \arcsin q^{2}

0 ⩽∣ \cos p ∣⩽ 1

the minimum is at p = (n - \frac{1}{2}) π

0 ⩽ 2 q

the minimum is at q = 0

trying we find absolute minima at

f (0, 0) = f (\pm π, 0) = 6

\Rightarrow minima at (x_{1} = 0 \lor x_{1} = \pm 1) \land x_{2} = 0

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