the second number is the reciprocal of the fist and the sm is a minimum.
this is my my work and i got stuck
x=first number
y=second number
so y=1/x
then what. i know the answer is 1 and 1.
also can you explain why the answer is 1 and 1 and not just one. it kind oes that to other problems too. like the answers to one of the problems was 55 and 55
by the way the opic is optimization problems
sum not sm
Ok, y= 1/x
xy= 1
sum= x + 1/x
take the derivative..
0= 1 -1 x^-2
x= sqrt 1 = 1
so y= 1
the min occurs at (x,y) of (1,1)
The 1 and 1 means x,y.
You have y(x) = x + 1/x
y'(x) = 1 -1/x2
Setting y' to 0 and solving you should get
-1/x2 = 1 or
x2 = 1
So x = +/-1
If you require the answer to be positive integers then the answer occurs at x=1, otherwise you'd need to look at the 3rd quadrant and there's no minimum there.
As to why you book lists both numbers, stating x=1 should be sufficient, but I think the authors are giving both numbers for completeness.
To solve the optimization problem given, where the second number is the reciprocal of the first and the sum is a minimum, you correctly set up the equations:
x = first number
y = second number
y = 1/x
To find the minimum value, you need to find the derivative of the sum and set it equal to zero.
sum = x + (1/x)
sum' = 1 - (1/x^2)
Now, set sum' equal to zero and solve for x:
0 = 1 - (1/x^2)
1/x^2 = 1
x^2 = 1
x = 1 or x = -1
Since you're looking for the minimum, you only consider positive values of x. In this case, x = 1.
Substituting x = 1 back into the equation for y, you get:
y = 1/1
y = 1
Therefore, the solution to the optimization problem is x = 1 and y = 1, or simply (1, 1).
As for why the answer is given as both 1 and 1 instead of just 1, it is likely for completeness. The authors want to give both values of x and y to fully represent the solution. Additionally, it may help to clarify that x and y are distinct values that are related to each other through the given equation y = 1/x. This pattern of listing both numbers can also be seen in other problems, such as the example you mentioned where the answer is given as 55 and 55. It serves the purpose of explicitly showing the values of both variables that satisfy the problem conditions.