Posted by bobby on Tuesday, December 16, 2008 at 7:52pm.
Find two numbers whose sum is 10 for which the sum of their squares is a minimum.
- Calculus - Reiny, Tuesday, December 16, 2008 at 7:55pm
two numbers : x and 10-x
S = x^2 + (10-x)^2
dS/dx = 2x - 2(10-x)
= 0 for max/min
x = 5
so the numbers are both 5
- Calculus - Damon, Tuesday, December 16, 2008 at 7:56pm
x and (10-x)
s = x^2 + (10-x)^2
s = x^2 + 100 -20x + x^2
s = 2 x^2 - 20 x + 100
s/2 = x^2 -10 x + 100 we can minimize half the sum easier than the whole sum
That is a parabola and you could find the vertex but since you said this was "calculus" we will take the derivative and set to zero.
0 = 2 x - 10
x = 5
10-x = 5
- Calculus - Damon, Tuesday, December 16, 2008 at 8:23pm
Interesting the answer is halfway between.
Say a sum of two numbers is s
We want to minimize the sum of squares of x^2 and (s-x)^2
sum = 2 x^2 -2sx
d sum/dx = 0 = 4 x -2s
x = s/2
so it works for any old sum, not just 10
Answer this Question
- Math - Minimizing Sum: Find two positive numbers whose sum is 100 and the sum of...
- math - Find two positive numbers whose sum is 50 such that the sum of their ...
- Math - Find two numbers whose difference is 16, and the sum of whose squares a ...
- maths - Find the two numbers whose differences is 8 and the sum of their ...
- calculus - Find two positive numbers whose product is 100 and whose sum is a ...
- Pre-Calculus - If the sum of two numbers is 1 and the sum of the cubes of the ...
- math - find two numbers whose product is N^2 and whose sum is a minimum
- math - Two non-negative numbers are chosen such that their sum is 30. Find the ...
- Math - Find two numbers whose ratio is 4 to 3 and are such that the sum of their...
- math - find two numbers whose sum is 25 and the sum of whose reciprocals is one ...