A company manufactures microchips. Use the revenue function R(x)=x(74−5x) and the cost function C(x)=120+15x to answer parts (A) through (D), where x is in millions of chips and R(x) and C(x) are in millions of dollars. Both functions have domain 1≤x≤20. (A) Form a profit function P, and graph R, C, and P in the same rectangular coordinate system. P(x)= (Type your answer in standard form.) (B) Discuss the relationship between the intersection points of the graphs of R and C and the x intercepts of P. The x-values of the intersection points of R and C are 1-greater compares 2- equal or 3-smaller compared to the x-intercepts of P. C) Find the x intercepts of P to the nearest thousand chips. Find the break-even points to the nearest thousand chips. The x-intercepts of P occur at x= million chips. (Use a comma to separate answers as needed. Round to three decimal places as needed.) Do the x-intercepts of the profit function indicate the break-even points? No Yes (D) Find the value of x (to the nearest thousand chips) that produces the maximum profit. Find the maximum profit (to the nearest thousand dollars), and compare it to the maximum revenue. The maximum profit occurs at x= million chips. (Round to three decimal places as needed.) The maximum profit is million dollars. (Round to three decimal places as needed.) The maximum revenue is million dollars. (Round to three decimal places as needed.)
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
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