compute the maximum profit for the profit function P(x)= x^3/3-9/2x^2+8x

as you know, max occurs where P' = 0

P'(x) = x^2 - 9x + 8 = (x-8)(x-1)

Since you know about the general shape of cubic curves, it should be clear which of these values provides a local maximum for P. Naturally, there is no global max for P.

Just plug in x=1 to get P(1)

thank you :) I got it

To compute the maximum profit for the profit function P(x) = x^3/3 - 9/2x^2 + 8x, we can follow these steps:

Step 1: Find the derivative of the profit function P(x) with respect to x.
Taking the derivative will give us the rate of change of the profit function at any given point.

P'(x) = d/dx (x^3/3 - 9/2x^2 + 8x)
= (1/3) * d/dx (x^3) - (9/2) * d/dx (x^2) + d/dx (8x)
= (1/3) * 3x^2 - (9/2) * 2x + 8
= x^2 - 9x + 8

Step 2: Set the derivative P'(x) equal to zero, and solve for x to find the critical points.
Setting the derivative equal to zero will give us the x-values at which the profit function has extreme points (maximum or minimum).

x^2 - 9x + 8 = 0

Factoring the equation:
(x - 1)(x - 8) = 0

Solving for x:
x - 1 = 0 -> x = 1
x - 8 = 0 -> x = 8

So, the critical points of the profit function are x = 1 and x = 8.

Step 3: Determine if each critical point is a maximum or minimum.
To determine whether each critical point is a maximum or minimum, we can use the second derivative test.

Taking the second derivative of the profit function P(x):
P''(x) = d^2/dx^2 (x^2 - 9x + 8)
= d/dx (2x - 9)
= 2

Since the second derivative P''(x) = 2 is positive for all x, this tells us that the function is concave up and has no points of inflection.

Using the second derivative test:
- If the second derivative is positive at a critical point, it means the function has a local minimum at that point.
- If the second derivative is negative at a critical point, it means the function has a local maximum at that point.

In our case, since P''(x) = 2 is positive, both critical points x = 1 and x = 8 correspond to local minimums.

Step 4: Calculate the profit at each critical point and compare.
To find the maximum profit, we need to calculate the profit function P(x) at each of the critical points and compare the results.

P(1) = (1^3/3) - (9/2 * 1^2) + (8 * 1)
= 1/3 - 9/2 + 8
= -19/6

P(8) = (8^3/3) - (9/2 * 8^2) + (8 * 8)
= 512/3 - 288/2 + 64
= 512/3 - 432/3 + 64
= 144/3
= 48

Therefore, the maximum profit for the given function P(x) = x^3/3 - 9/2x^2 + 8x is 48, which occurs at x = 8.