Question

If y=3x-7, x>0, what is the minimum product of x^2y?

Answer 1

please, note that, your function is:
\[f(x)=x ^{2}(3x-7)\]
you have to explicit it, first step

Answer 2

This is an optimization problem. Let P represent the product. Then\[P =x ^{2}y\]If\[y =3x -7\]Then\[P =x ^{2}(3x -7)\]Now you should know how to use Calculus to solve the problem from here.

Answer 3

please use this theore:
"A function f(x) is an increasing function if and only if the first derivative f'(x) is positive".
I other words you have to find points x, at which your first derivative is positive

Answer 4

oops... theorem...

Answer 5

hmm x>0 can be a tricky constraint

Answer 6

Is the graph of the product function, Thus you can see that the minimum product will be when 0 < x < 7/3.

Answer 7

Now use calculus to find the x (the critical number).

Answer 8

@calculusfunctions please note that the function of @chodefet is not definite for x<0

Answer 9

Yes I know that but thank you for your input anyway. I just drew the general graph. However if you you observe properly, you'll see I wrote that 0 < x < 7/3.

Answer 10

@calculusfunctions you are right!

Answer 11

Well @chodefet doesn't seem to be replying and so @Michele_Laino hopefully you can guide him without writing the solution for him because I will be logging out now.

Answer 12

Let P=x²y, then

P = x²y
..= x²(3x-7)
..= 3x³ - 7x²

Answer 13

@calculusfunctions ok! I try!

Answer 14

Take the first derivative
P'= 9x² - 14x
..= x(9x - 14)
..= 0 for x=0 and x=14/9

Answer 15

okay so from here, do i just go through the process of derivative testing to find the minimum?
thanks guys!

Answer 16

Since x must be greater than zero,
we are left to consider only x=14/9.

The question now is:
Does P have a minimum at x=14/9?

Take the second derivative of P
P''= 18x - 14

Answer 17

thanks guys! i really appreciate the help!

Answer 18

For x=14/9, P''= 18(14/9)-14 = 14 which is positive.
Therefore P''>0 at x=14/9.
So P is concave up at x=14/9 and thus

P has a relative minimum at x=14/9

Answer 19

Now, for x=14/9
y = 3x-7
y = 3(14/9)-7
y = -7/3

So, the minimum for x²y is
P = x²y
P = (14/9)²(-7/3)
p = -5.646

Answer 20

@chodefet