The profit P (in thousands of dollars) for a company that spends an amount s (in thousands of dollars) on advertising is P = − 1/10s^3 + 15s^2 + 400.
Find the point of diminishing returns.
define "point of diminishing returns", not familiar with this kind of math
point of diminishing returns is the point at which the growth rate of the earnings function begins to decline.
To find the point of diminishing returns, we need to find the value of s where the rate of increase in profit starts to decline. This occurs when the derivative of the profit function equals zero.
Step 1: Find the derivative of the profit function.
To find the derivative of the profit function, we differentiate each term of the function separately. The derivative of each term of the function P = -1/10s^3 + 15s^2 + 400 is as follows:
dP/ds = d/ds (-1/10s^3) + d/ds (15s^2) + d/ds (400)
= -3/10s^2 + 30s
Step 2: Set the derivative equal to zero and solve for s.
To find the point of diminishing returns, we equate the derivative to zero and solve for s:
-3/10s^2 + 30s = 0
We can factor out s from the equation:
s(-3/10s + 30) = 0
Setting each factor equal to zero individually:
s = 0
-3/10s + 30 = 0
For s = 0, profit is zero, which does not correspond to a point of diminishing returns.
Solving for s in the second equation:
-3/10s + 30 = 0
-3/10s = -30
s = (-30)(10)/(-3)
s = 100
Thus, the point of diminishing returns occurs when the company spends $100,000 on advertising.
Note: It is important to consider the context of the problem and ensure the result is reasonable. Diminishing returns occur when the rate of increase in profit starts to decline, indicating that additional spending on advertising may not yield significant increases in profit.