when sum of any quantities is constant, there product is maximum when they are equal.
example. if 3x+5y=15.
find maximum value of x^2*y^3.
here 3x+5y=15 => 3x/2 + 3x/2 + 5y/3 + 5y/3 + 5y/3 = 15.--------------1as I said, when sum of any quantities is constant, there product is maximum when they are equal. :beers
here sum is constant. so when 3x/2 = 5y/3. we get maximum value of x^2*y^3. taking 3x/2 = 5y/3 putting it in 1, => 5(3x/2) = 15. =>x=2. and y = 9/5. answer is 2^2*(9/5)^3. :beers
generalizing it, how to find maximum value of x^m*y^n where ax+by=P. a,b,x,y>0 x^m*y^n is maximum when ax/m = by/ n = p/m+n 4. when the product of any quantity is constant, sum of the all the quantity is minimum, when they are equal. xy^3 = 64.
find minimum value of x+12y. we need to adjust x+12y,accordingly. x+12y = x+ (12y/3)*3now, x*(12y/3)^3= 64 *64 ( coz xy^3 = 64)-----------1 the product is constant. so the sum of the quantities will be minimum when quantities are equal. take x= 12y/3 putting it in 1, we get x= 8 =>12y/3 = 8, y = 2. minimum value of x+12y = 8+24 = 32.
generalizing it, how to find minimum value of ax+by where x^m*y^n=P a,b,x,y>0 ax+by is minimum when ax/m = by/n
In a solid figure no. of faces + vertices = no. of edges+2
i.e. F+V=E+2
If a larger cube painted by a color is broken into n smaller cubes, then no. of cubes having
3 faces painted = 8
2 faces painted = 12(n-2)
1 face painted = 6(n-2)^2
0 faces painted = (n-2)^3
There are n non-overlapping identical triangles, then at max how many bounded regions these intersecting triangle can form.
The distance b/w P&Q is 100m and the speeds of A and B 20m/s and 30m/s respectively.Initially A and B are at P.The distance B/w Q & the point of 3rd meeting?