If y=3x-7, x is greater than 0, what is the minimum product of x^2y?
A. -5.646
B. 0
C. 1.556
D. 2.813
E. 4.841
Substitute y = 3x - 7 to x^2*y:
x^2 * (3x - 7)
3x^3 - 7x^2
Then differentiate with respect to x, and equate to zero (since we're looking for maximum/minimum):
9x^2 - 14x = 0
x(9x - 14) = 0
x = 0
x = 14/9
Note that if x = 0, the minimum product of x^2 * y is zero.
But if x = 14/9, substituting this back to get y,
y = 3(14/9) - 7
y = 14/3 - 7
y = -7/3
Thus,
x^2 * y
= (14/9)^2 * (-7/3)
= -5.6461 (which is smaller than zero)
hope this helps~ `u`
Well, well, it looks like x is greater than 0, but that doesn't mean we can't have a little fun with this question! Let's plug in some numbers and see what happens.
If x = 1, then y = 3(1) - 7 = -4. So, x^2y = 1^2(-4) = -4.
If x = 2, then y = 3(2) - 7 = -1. So, x^2y = 2^2(-1) = -4.
If x = 3, then y = 3(3) - 7 = 2. So, x^2y = 3^2(2) = 18.
Hmm... it seems like we're going up and down like a yo-yo! But we're trying to find the minimum product here, so let's keep going:
If x = 4, then y = 3(4) - 7 = 5. So, x^2y = 4^2(5) = 80.
If x = 5, then y = 3(5) - 7 = 8. So, x^2y = 5^2(8) = 200.
Aha! We've hit a turning point! As x increases beyond 5, the value of x^2y keeps getting bigger. Therefore, the minimum product occurs when x = 5.
Plugging x = 5 into our equation, we find y = 3(5) - 7 = 8. So, x^2y = 5^2(8) = 200.
Wow, looks like the minimum product of x^2y is 200! But that wasn't one of the options given... So, it seems like there's a typo in the answer choices. I guess we'll have to wait for the circus to fix that one!
To find the minimum product of x^2y, we need to substitute the value of y in terms of x and then minimize the expression.
Given, y = 3x - 7
Now, we can substitute this value of y in x^2y:
x^2 * y = x^2 * (3x - 7)
= 3x^3 - 7x^2
To minimize this expression, we need to find the critical points. We can do this by taking the derivative of the expression and setting it equal to zero:
d/dx (3x^3 - 7x^2) = 9x^2 - 14x = 0
Factoring out x:
x(9x - 14) = 0
From this, we can see that the critical points are x = 0 and x = 14/9.
To find the minimum product, we need to evaluate the expression at these critical points and determine which one gives the minimum value:
When x = 0:
3x^3 - 7x^2 = 3(0)^3 - 7(0)^2 = 0
When x = 14/9:
3x^3 - 7x^2 = 3(14/9)^3 - 7(14/9)^2 ≈ 4.841
Comparing the two values, we can see that the minimum product occurs when x = 14/9.
Therefore, the correct answer is E. 4.841.
To find the minimum product of x^2y, we need to minimize both x and y.
Given that x is greater than 0 and the equation y = 3x - 7, we need to find the minimum value of x and substitute it into the equation to find the corresponding minimum value of y.
To minimize y, we want to minimize x. Since x is greater than 0, the smallest possible value for x is 0.
Now, substituting x = 0 into the equation y = 3x - 7:
y = 3(0) - 7
y = -7
So when x = 0, y = -7.
To find the minimum product of x^2y, we simply calculate:
(0)^2 * (-7) = 0
Therefore, the minimum product of x^2y is 0.
Therefore, the correct answer is option B.