Cauchy-Schwarz Training

The biggest real value of the expression $\cos{x} \cdot ( 2\sin{x} + 4\cos{x})$ can be written as $a+ \sqrt{b}$ , where $a$ and $b$ are positive square-free integers.

The smallest real value of the expression $ay^2 + \dfrac{b}{y^2}$ can be written as $c \cdot \sqrt{d}$ , where $c$ and $d$ are positive square-free integers. Restrict $x$ and $y$ to the real numbers.

Show that $ab = bc = d$ .
Find the monic polynomial $P(x)$ of fourth degree that has $a,b,c,d$ as its roots.
Find $P(x)$ 's smallest real value.

#Algebra #Calculus #CauchySchwarzInequality #MinimaAndMaxima

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

The first expression can be reduced to $\sin 2x+2\cos 2x+2$ .

$(\sin 2x+2\cos 2x)^{2} \leq (\sin^{2} 2x+ \cos^{2} 2x)(1+4)=5$ hence the maximum value is $2+\sqrt {5}$ . Equality can occur when $\tan x=0.5$ .

Now $c, d$ can be found by AM-GM. However, since Cauchy-Schwarz needs to be used, we do it in a slightly different way.

$(2y^{2}+\frac {5}{y^{2}})(\frac {5}{y^{2}}+2y^{2}) \geq (\sqrt {10}+\sqrt {10})^{2}=40$ hence $2y^{2}+\frac {5}{y^{2}} \geq \sqrt {40}=2\sqrt {10}$ .

Hence $a=2, b=5, c=2, d=10$ .

Clearly $2×5=5×2=10$ .
$P (x)=(x-2)^{2}(x-5)(x-10)$
Using Cauchy-Schwarz rather than calculus is difficult, but here's a solution:

By Cauchy-Schwarz, $(a^{2}+b^{2})(b^{2}+a^{2}) \geq (ab+ba)^{2}$ hence $a^{2}+b^{2} \geq 2ab$ for reals $a, b$ .

Thus for all positive reals $a, b, c, d$ , $a^{4}+b^{4}+c^{4}+d^{4} \geq 2a^{2} b^{2}+2c^{2} d^{2} \geq 4abcd$ by applying the above twice. We will use this fact.

Clearly the polynomial can have negative values. Just fit in x=7.

Now the only way to get negative values is to have $(x-5)(x-10)$ negative, or $5 < x <10$ . Hence $x-2, x-5, 10-x$ are positive. So an easier problem would be to find the maximum of $(x-2)(x-2)(x-5)(10-x)$ .

Now, using the fact above, $r(x-2)r (x-2)(1-2r)(x-5)(10-x) \leq (\frac {r(x-2)+r (x-2)+(1-2r)(x-5)+10-x}{4})^{4}=(\frac {6r+5}{4})^{4}$ . So the maximum, if equality can occur, is $\frac {(6r+5)^{4}}{256r^{2}(1-2r)}$ .

In particular, when $r=\frac {21-\sqrt {321}}{12}$ , we get the maximum as that is the only case equality can occur. (Remember that the corresponding terms in Cauchy-Schwarz have to be related by a common ratio for equality.) The minimum is the negative of the maximum and is approximately -223. Equality can occur at $x=\frac {64-4\sqrt {321}}{17-\sqrt {321}}$ .

Joel Tan - 6 years, 7 months ago

Can you explain adequately to your friends:

a) Why is the "maximum" $\dfrac{(6r+5)^4}{256r^2 (1-2r)}$ ? How did attain it?

b) Why is $r = \dfrac{21 - \sqrt{321}}{12}$ ?

Guilherme Dela Corte - 6 years, 6 months ago

a) it is by rearranging the terms in the inequality. b) By Cauchy, equality can occur only when r(x-2)=(1-2r)(x-5)=10-x. Solving gives r.

Joel Tan - 6 years, 6 months ago

Can anyone tell me about Cauchy-Schwarz inequality? I know the AM-GM inequality but not this. Please illustrate some simple examples too (I am new to this inequality.).

Thanks in advance!

Ninad Akolekar - 6 years, 7 months ago

If you type in Cauchy-Schwarz in the search box there would be a featured community post on Cauchy-Schwarz. There's a lot of information there.

Joel Tan - 6 years, 6 months ago

Challenge: Can the third question be answered by Cauchy-Schwarz?

Guilherme Dela Corte - 6 years, 7 months ago

Waht's the question which has no answer ?

Hazem Gamal - 6 years, 6 months ago

All questions have answers! However, #1 is a proof, and thus has not a closed-form value to submit.

Guilherme Dela Corte - 6 years, 6 months ago

Write a comment or ask a question...Find K so that U and V are orthogonal where U=(2,3K,-4,1,-5) and V=(6,-1,3,7,2K)

Sunny Bassey - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$