JEEAdvanced
Let x, y, z be positive real numbers with x + y + z = 12. Find the…
Question
Let x, y, z be positive real numbers with x + y + z = 12. Find the minimum possible value of 9/x + 16/y + 25/z.
✓ Verified answer: 12checked by our engine — not a guess
Step-by-step solution
By the Cauchy-Schwarz inequality in Engel (Titu) form, 9/x + 16/y + 25/z = 3^2/x + 4^2/y + 5^2/z >= (3+4+5)^2/(x+y+z) = 144/12 = 12.
Equality holds when 3/x = 4/y = 5/z, i.e.
x:y:z = 3:4:5, giving x=3, y=4, z=5 (sum 12), which lies in the feasible region.
Hence the minimum is 12.
Final answer12
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Maxima Minima solutions
A straight line of negative slope passes through the point (3, 4) and…JEE · MathLet x, y, z be positive real numbers with x + y + z = 6. Find the…JEE · MathFind the shortest distance from the point (0, 17/4) to the parabola y…JEE · MathA point P moves along the x-axis. Let L(P) = PA + PB where A = (0, 2)…JEE · MathA rectangle has its base on the x-axis and its two upper vertices on…JEE · MathA right circular cone is inscribed in a sphere of radius R so that…JEE · MathFind the minimum value of f(x) = 1·|x - 1| + 2·|x - 2| + 3·|x - 3| +…JEE · MathLet m(a) = max over x in [-1, 1] of |x^3 - a·x|. As a ranges over all…JEE · Math