Question

Find the point of diminishing retums (x,y) for the given function, where R(x) represonts revenue (n thousands of dollarn) and x fepresents the amcent spent on adintlining (in thousands ol dolan) R(x)=−0.2x 3 +4.3x 2 +4x,05x≤22 The point of diminishing returns occurs at (Type an ordered pai. Do not round unti the fnal answec. Then round to two decimal places as needed)

Answer 1

Given the revenue function R(x) = -0.2x^3 + 4.3x^2 + 4x, the point of diminishing returns—that is, the maximum point on the graph of the function—is found by differentiating the function to get its derivative, finding the areas where the derivative equals zero (these are potential maximum or minimum points), and then checking these solutions.

Let's start by finding the derivative of the revenue function. This will give us the rate of change of the revenue with respect to increases in the ad spend.

After calculating the derivative and the second derivative of our function, we narrow down our search for the maximum to the range 0.5 ≤ x ≤ 22 given that x represents our monetary units spent and we are only considering this particular range.
Through our calculations, we find the second derivative of R(x) achieves a zero at x = 7.17 approximately.

Remember, the second derivative test tells us that a function achieves a local maximum if the second derivative is negative at that point. This means that at x = 7.17,the function R(x) is at a maximum.
Therefore, we substitute x = 7.17 into our original function R(x) to find the corresponding y-value. That gives us y = 175.90 approximately.

So, the point of diminishing returns is at approximately (7.17, 175.90), with the understanding that the amounts are in thousands.