Mathematics

Mathematics

Problem

A company’s profit (in thousands of rupees) from selling xx hundred units of a product is given by P(x)=100xx+4−5xP(x)=\dfrac{100x}{x+4}-5x, where x>0x>0. a. (a) Find the interval of xx for which the profit is increasing. b. (b) Determine the number of units that give the maximum profit.
Intuitive explanation

The key is recognizing this as an optimization problem where you need to find the maximum profit by balancing revenue against costs. Take the derivative of P(x)P(x), set it equal to zero, and solve for xx—this finds the critical point where profit stops increasing. Then verify it's a maximum (not a minimum) using the second derivative test or by checking the sign of P′(x)P'(x) around your critical point.

Intuitive explanation

The key is recognizing this as an optimization problem where you need to find the maximum profit by balancing revenue against costs. Take the derivative of P(x)P(x), set it equal to zero, and solve for xx—this finds the critical point where profit stops increasing. Then verify it's a maximum (not a minimum) using the second derivative test or by checking the sign of P′(x)P'(x) around your critical point.

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