For each of the following graphs, use interval notation to express the values of a on which the graph is: i. the value of the function is increasing ii, the value of the function is decreasing iii. the rate of change of y with respect to x is increasing iv, the rate of change of y with respect to a is decreasing Write DNE if there are no intervals of r which satisfy the description. The graphs, from left to right, are A,B, and C. 6- 5+ -3 2 2- A. i. Function is increasing: (-2,2) Preview ii. Function is decreasing: (-o0,-2)U(2,00) Preview Preview iii. Rate of change of y with respect to a is increasing: iv. Rate of change of y with respect to r is decreasing: Preview B. i. Function is increasing: (-00,5) Preview ii. Function is decreasing: (.5,0o) Preview iii. Rate of change of y with respect to z is increasing: Preview iv. Rate of change of y with respect to z is decreasing: Preview C. i. Function is increasing: (-00,-JU[-5,2.5) * Preview ii. Function is decreasing: (-3,-1)U(2.5,00) Preview iii. Rate of change of y with respect to r is increasing: Preview iv. Rate of change of y with respect to x is decreasing: Preview
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.
For each of the following graphs, use interval notation to express the values of x on which the graph is:
- the value of the function is increasing
- the value of the function is decreasing
- the rate of change of y with respect to x is increasing
- the rate of change of y with respect to x is decreasing
Write DNE if there are no intervals of x which satisfy the description.
The graphs, from left to right, are A,B, and C.
- Function is increasing:
- Function is decreasing:
- Rate of change of y with respect to x is increasing:
- Rate of change of y with respect to x is decreasing:
- Function is increasing:
- Function is decreasing:
- Rate of change of y with respect to x is increasing:
- Rate of change of y with respect to x is decreasing:
- Function is increasing:
- Function is decreasing:
- Rate of change of y with respect to xx is increasing:
- Rate of change of y with respect to xx is decreasing:
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