Difficult Calculus Question

[JackSonSmith]

For the variables x, y and z, it is known that  z = (x^2) * (y)  and  2x + y = 50.

Find the maximum value of z if:

 0<=x<=25 (greater than or equal to)

This question is a part of "Applications to maximum and minimum and rate problems"

[plato]

From the second equation, y=50-2x
Substitute into the first equation
z=x²(50-2x)

Now differentiate and find where gradient = 0.
Then check if this point is a maximum or a minimum.

[JackSonSmith]

Thankyou

[keltingmeith]

From the second equation, y=50-2x
Substitute into the first equation
z=x²(50-2x)

Now differentiate and find where gradient = 0.
Then check if this point is a maximum or a minimum.

Plato has forgotten one important bit - while this method will tell you about local maxima/minima, it won't tell you about the values at end-points. So, you will also need to confirm the end-points of the equation don't rise above/below these stationery points.

[plato]

Thanks EulerFan101. That was careless of me.

[plato]

On reflection, the domain was given as [0, 25]
The cubic z=x²(50-2x) is a negative cubic with a local mimimum at (0, 0) and another x intercept at (25, 0)

Within the given domain, this cubic is positive with a local maximum that does give the maximum value of z.

Regardless, I should have taken the domain into account earlier.