The weekly profit P for a widget producer is a function of the number n of widgets sold. The formula is as follows, where P is measured in thousands of dollars, n is measured in thousands of widgets, and the formula is valid up to a level of 12 thousand widgets sold. P = −1 + 2.5n − 0.2n2 (b) Calculate P(0). P(0) = $ thousand Explain in practical terms what your answer means. The company will gain profit if they produce 0 widgets.The company will lose profit if they produce 0 widgets. The company will gain profit in week 0.The company will lose profit in week 0. (c) At what sales level is the profit as large as possible? (Use the graph to find the value. Round your answer to the nearest whol
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.
The weekly profit P for a widget producer is a function of the number n of widgets sold. The formula is as follows, where P is measured in thousands of dollars, n is measured in thousands of widgets, and the formula is valid up to a level of 12 thousand widgets sold.
P(0) = $ thousand
Explain in practical terms what your answer means.
(c) At what sales level is the profit as large as possible? (Use the graph to find the value. Round your answer to the nearest whole number.)
n = thousand widgets
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