The following graph shows a rough approximation of historical and projected median home prices for a country for the period 2000-2024. C(1) 140 120 100 80 60 40 20 24 O 3 6 9 12 15 19 21 Here, t is time in years since the start of 2000, and C(t) is the median home price in thousands of dollars. The locations of stationary points and points of inflection are indicated on the graph. Analyze the graph's important features, and interpret each feature in terms of the median home price. The median home price was $ thousand at the start of 2000 (t = 0). The median home price has two low points; first in the year and again in the year when it stood at $ thousand; the median home price peaked at the start of the year at $ thousand. The median home price was decreasing most rapidly at the start of the year when it was $ thousand, and increasing most rapidly at the start of the year when it was $ thousand. Assuming that the trend shown in the graph continued indefinitely, the median home price would approach a value of $ thousand in the long term.
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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