P(x) = –2x^3+ 36x^2– 100x + 200Find the point of diminishing returns for the profit function P(x) which represents profit in thousands of dollars and xrepresents the amount spent on advertising...

P(x) = –2x^3+ 36x^2– 100x + 200

Find the point of diminishing returns for the profit function P(x) which represents profit in thousands of dollars and x
represents the amount spent on advertising in thousands of dollars. (The point of diminishing returns is the point at which the
rate of growth of the profit function begins to decline, i.e. the point of inflection.)

Asked on by jmg5639

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to solve the second derivative equation `P''(x)=0`  to find the inflection point.

Hence, you should evaluate the first derivative since `P''(x) = (P'(x))'`  such that:

`P'(x) = (-2x^3 + 36x^2 - 100x + 200)'`

`P'(x) = -6x^2 + 72x - 100`

You may evaluate `P''(x)`  such that:

`P''(x) = (-6x^2 + 72x - 100)'`

`P''(x) = -12x + 72`

Solving the equation `P''(x) = 0`  you will find the inflection point such that:

`-12x + 72 = 0 =gt -12 x = -72 `

`x = 6 =gt P(6) = -432 + 1296- 600 + 200 = 464`

Hence, evaluating the point of diminishing returns of profit function yields `(6,464).`

Sources:

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