Question

Profit Function for Producing Thermometers The Mexican subsidiary of ThermoMaster manufactures an indoor-outdoor thermometer. Management estimates that the profit (in dollars) realizable by the company for the manufacture and sale of x units of thermometers each week is represented by the function below, where x ≥ 0. Find the interval where the profit function P is increasing and the interval where P is decreasing. (Enter your answer using interval notation.) P(x) = −0.004x2 + 6x − 5,000 Increasing: Decreasing:

Answer 1

Answer:

Increasing: $(0, 750)$

Decreasing: $(750, \infty)$

Step-by-step explanation:

Critical points:

The critical points of a function f(x) are the values of x for which:

$f'(x) = 0$

For any value of x, if f'(x) > 0, the function is increasing. Otherwise, if f'(x) < 0, the function is decreasing.

The critical points help us find these intervals.

In this question:

$P(x) = -0.004x^{2} + 6x - 5000$

So

$P'(x) = -0.008x + 6$

Critical point:

$P'(x) = 0$

$-0.008x + 6 = 0$

$0.008x = 6$

$x = \frac{6}{0.008}$

$x = 750$

We have two intervals:

(0, 750) and $(750, \infty)$

(0, 750)

Will find P'(x) when x = 1

$P'(x) = -0.008x + 6 = -0.008*1 + 6 = 5.992$

Positive, so increasing.

Interval $(750, \infty)$

Will find P'(x) when x = 800

$P'(x) = -0.008x + 6 = -0.008*800 + 6 = -0.4$

Negative, then decreasing.

Answer:

Increasing: $(0, 750)$

Decreasing: $(750, \infty)$