If log10x3+y3-log10x2+y2-xy≤2, the maximum value of xy, for all x≥0,y≥0 is
2500
3000
1200
3500
Explanation for the correct answer:
Given: log10x3+y3-log10x2+y2-xy≤2
⇒log10x3+y3x2+y2-xy≤2[∵loga-logb=logab]⇒log10(x+y)(x2+y2-xy)x2+y2-xy≤2[∵(x3+y3)=(x+y)(x2+y2-xy)]⇒log10x+y≤2⇒(x+y)≤102[∵logab=c⇒b=ac]⇒x+y≤100
We know that AM≥GM where AM is Arithmetic mean and GM is Geometric mean
⇒xy≤x+y2⇒xy≤1002[∵(x+y)≤100]⇒xy≤50⇒xy≤2500
Maximum value of xy=2500
Hence option(A) i.e. 2500 is correct.
If log10(x3+y3)-log10(x2+y2-xy)≤2, then the maximum value ofxy, for all x≥0,y≥0, is
If log10(x3+y3)–log10(x2+y2–xy)≤2, then the maximum value of xy for all x≥0,y≥0 is: