Given that xy = 3/2 and both x and y are nonnegative real numbers, find the minimum value of 10x + 3y/5
y = 3/(2x)
so we want the minimum of
10x + 3/5 * 3/(2x)
= 10x + 9/(10x)
take the derivative and you have
10 - 9/(10x^2)
set that to zero and you get
100x^2 - 9 = 0
(10x-3)(10x+3) = 0
x = ±3/10
So, at x = 3/10, y = 5
and 10x+3y/5 = 3+3 = 6
thanks for the help!
To find the minimum value of 10x + 3y/5, we can use the given equation xy = 3/2 to express one variable in terms of the other.
Since both x and y are nonnegative real numbers, we can solve for one variable in terms of the other. Let's solve for y in terms of x:
xy = 3/2
y = (3/2)/x
y = 3/(2x)
Now substitute this expression for y into the expression 10x + 3y/5:
10x + 3y/5 = 10x + 3(3/(2x))/5
Simplify this expression:
10x + 3(3/(2x))/5
= 10x + 9/(2x*5)
= 10x + 9/(10x)
Now, we can find the minimum value of this expression by finding its derivative with respect to x and setting it equal to zero. Let's differentiate the expression:
(d/dx)(10x + 9/(10x)) = 10 - 9/(10x^2)
Setting the derivative equal to zero:
10 - 9/(10x^2) = 0
Multiplying both sides by 10x^2, we get:
10x^2 - 9 = 0
Adding 9 on both sides:
10x^2 = 9
Dividing both sides by 10:
x^2 = 9/10
Taking the square root of both sides:
x = ±√(9/10)
Since x and y are nonnegative real numbers, we consider only the positive square root:
x = √(9/10)
Now substitute this value of x into the expression for y:
y = 3/(2x)
y = 3/(2 * √(9/10))
y = 3/(2 * (3/√10))
y = 3/(6/√10)
y = √10/2
So, the values of x and y that minimize the expression 10x + 3y/5 are x = √(9/10) and y = √10/2.
To find the minimum value of the expression, substitute these values into the expression:
10x + 3y/5 = 10(√(9/10)) + 3(√10/2)/5
= 10√9/√10 + 3√10/10
= 10/√10 + 3√10/10
= √10 + 3√10/10
= (4√10)/10
= 2√10/5
Therefore, the minimum value of 10x + 3y/5 is 2√10/5.
To find the minimum value of the expression 10x + 3y/5, we need to use the given condition xy = 3/2 and optimize the expression with respect to x and y.
Let's start by isolating y in terms of x from the given condition xy = 3/2. Divide both sides of the equation by x:
y = (3/2) / x
Now, substitute this value of y into the expression 10x + 3y/5:
10x + 3((3/2) / x)/5
10x + 9x/(10x)
(100x^2 + 9x) / (10x)
To find the minimum value, we need to find the critical points of the expression. These are the values of x for which the denominator is equal to zero (since division by zero is undefined).
Setting the denominator equal to zero:
10x = 0
x = 0
Note that x = 0 is the critical point.
Now, to check whether this critical point is a minimum or maximum, we can take the second derivative. However, since the expression 10x + 3y/5 has a linear term only, there is no curve to analyze, and the minimum value occurs at the critical point x = 0.
Therefore, the minimum value of 10x + 3y/5 is achieved when x = 0, and the minimum value is 10(0) + 3(0)/5 = 0.