The daily sales, S, in thousands of dollars, is a function of the money, x, spent on advertising (in thousands of dollars) according to: S(x)= -x^3+9x^2+6/6. Find the x-coordinate of the point of diminishing returns. How much money is spent on advertising when you reach the point of diminishing returns?
Wouln't the point at which S'=0 change the sign of the returns?
To find the x-coordinate of the point of diminishing returns, we need to find the value of x when the derivative of the sales function, S'(x), equals 0.
Let's begin by finding the derivative of S(x) with respect to x. The derivative of -x^3 + 9x^2 + 6/6 is obtained by differentiating each term separately:
S'(x) = d/dx (-x^3) + d/dx (9x^2) + d/dx (6/6)
Differentiating each term:
S'(x) = -3x^2 + 18x + 0
Setting S'(x) equal to 0:
-3x^2 + 18x = 0
Factoring out common terms:
3x(-x + 6) = 0
Using the zero product property, we can set each factor equal to zero:
3x = 0 or -x + 6 = 0
Solving for x in each equation:
1) 3x = 0
x = 0
2) -x + 6 = 0
x = 6
Now, we have two possible x-coordinates for the point of diminishing returns: x = 0 and x = 6.
To determine which one represents the point of diminishing returns, we need to check the second derivative of the sales function (S''(x)) at each x-coordinate. If the second derivative is positive, it means that the function is concave up and represents a point of diminishing returns. If it is negative, it represents a point of increasing returns.
To find the second derivative, we differentiate S'(x) again:
S''(x) = d/dx (-3x^2 + 18x)
Differentiating each term:
S''(x) = -6x + 18
Now, substitute x = 0 into S''(x):
S''(0) = -6(0) + 18
S''(0) = 18
Since S''(0) = 18 is positive, it indicates a concave-up shape and represents a point of diminishing returns.
Now, substitute x = 6 into S''(x):
S''(6) = -6(6) + 18
S''(6) = -6
Since S''(6) = -6 is negative, it indicates a concave-down shape and represents a point of increasing returns.
Therefore, the x-coordinate of the point of diminishing returns is x = 0. This means that when no money is spent on advertising (x = 0), the sales start to have diminishing returns.
To determine how much money is spent on advertising at the point of diminishing returns, we substitute x = 0 back into the sales function:
S(0) = -0^3 + 9(0)^2 + 6/6
S(0) = 0 + 0 + 1
S(0) = 1
So, when you reach the point of diminishing returns, $1,000 (in thousands) is spent on advertising.