If log 10 x 3- y 3-log 10 x 2- xy - y 2 - 2 - x -0- y -0- then the maximum value of x - y isA- 50B- 1200C- 2500D- 100

For the inequality to be defined, x^3+y^3 gt;0 and x^2 xy+y^2 gt;0 log_10(x^3+y^3) log_10(x^2 xy+y^2)≤ 2 ⇒log_10 ( x^3+y^3x^2 xy+y^2 )≤ 2 ⇒log_10(x+y) ≤ ...

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Byju's Answer

Standard XII

Mathematics

Logarithmic Inequalities

If log 10 x 3...

Question

If ${log}_{10} (x^{3} + y^{3}) - {log}_{10} (x^{2} - x y + y^{2}) \leq 2 \forall x > 0, y > 0$ , then the maximum value of $x + y$ is

50

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100

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2500

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1200

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Solution

The correct option is B $100$
For the inequality to be defined,
$x^{3} + y^{3} > 0 and x^{2} - x y + y^{2} > 0$

${log}_{10} (x^{3} + y^{3}) - {log}_{10} (x^{2} - x y + y^{2}) \leq 2$
$\Rightarrow {log}_{10} (\frac{x^{3} + y^{3}}{x^{2} - x y + y^{2}}) \leq 2 \Rightarrow {log}_{10} (x + y) \leq 2 \Rightarrow x + y \leq 100$
$∴$ Maximum value of $x + y$ is $100.$

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If log10(x3+y3)−log10(x2−xy+y2)≤2 ∀ x>0, y>0, then the maximum value of x+y is

The correct option is B 100For the inequality to be defined, x3+y3>0 and x2−xy+y2>0 log10(x3+y3)−log10(x2−xy+y2)≤2 ⇒log10(x3+y3x2−xy+y2)≤2⇒log10(x+y)≤2⇒x+y≤100 ∴ Maximum value of x+y is 100.

If ${log}_{10} (x^{3} + y^{3}) - {log}_{10} (x^{2} - x y + y^{2}) \leq 2 \forall x > 0, y > 0$ , then the maximum value of $x + y$ is