Use y - (x² – 6x)² and its derivative Y - 4x(x – 3)(x – 6) to find each of the following. dx Find the critical values. (Enter your answers as a comma-separated list.) Find the critical points. (x, y) = ( (smallest x-value) (x, v) = (| (x, y) = ( (largest x-value) Find the intervals on which the function is increasing. (Enter your answer using interval notation.) Find the intervals on which the function is decreasing. (Enter your answer using interval notation.) Find the relative maxima, relative minima, and horizontal points of inflection. (If an answer does not exist, enter DNE.) relative maxima relative minima (smaller x-value) relative minima (larger x-value) (x, V) = ( horizontal points of inflection Sketch the graph of the function. 200 200 100 100 -10 -5 10 -10 10 -100 -100 -200 -200 y 200 200 100 -10 -5 10 -10 -5 10 -100 -1bo- -200- -200-
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.
![Use \( y = (x^2 - 6x)^2 \) and its derivative \(\frac{dy}{dx} = 4x(x - 3)(x - 6)\) to find each of the following.
### Find the Critical Values
(Enter your answers as a comma-separated list.)
- \( x = \) [ ]
### Find the Critical Points
- \((x, y) =\) [ ] (smallest x-value)
- \((x, y) =\) [ ]
- \((x, y) =\) [ ] (largest x-value)
### Find the Intervals on which the Function is Increasing
(Enter your answer using interval notation.)
- [ ]
### Find the Intervals on which the Function is Decreasing
(Enter your answer using interval notation.)
- [ ]
### Find the Relative Maxima, Relative Minima, and Horizontal Points of Inflection
(If an answer does not exist, enter DNE.)
- Relative Maxima \((x, y) =\) [ ]
- Relative Minima \((x, y) =\) [ ] (smaller x-value)
- Relative Minima \((x, y) =\) [ ] (larger x-value)
- Horizontal Points of Inflection \((x, y) =\) [ ]
### Sketch the Graph of the Function
Four graphs are provided, all representing sketches of the function. Each graph has coordinates labeled with 'x' and 'y' axes ranging from approximately -10 to 10 on the x-axis and from -200 to 200 on the y-axis. The function appears to have multiple turning points. The options are marked as "(i)" at the bottom left of each graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F519381ce-35b2-452f-b130-24c2d7c48980%2F14ff196e-eee8-4265-aa61-9a057e5ffcab%2Fk8cbei_processed.jpeg&w=3840&q=75)
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