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.
What must be done to a function’s equation so that its graph is stretched vertically? Explain.
Suppose that the function is say y = f(x) .
To find what must be done to a function's so that its graph is stretched vertically.
When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in the relation to graph of the original function.
If the constant is greater than 1, we get a vertical stretch .
If the constant is between 0 to 1, we get a vertical compression.
Suppose the given function is f (x) , a new function g ( x) = c f(x) , where c is a constant is a vertical stretch if c > 1.
If c < 0 , then there will be combination of a vertical stretch or compression with vertical reflection.
Example:
let f(x) = sin x and stretch the graph of f(x) vertically by 2 units.
Multiply the given function by 2 units.
Then,
The new transformation function is g(x) = 2. f(x) = 2.sin x .
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