Analyze the polynomial function f(x)= (x + 4)(x- 5)(x + 6) using parts (a) through (e). a) Determine the end behavior of the graph of the function. The graph of f behaves like y = for large values of x|. b) Find the x- and y-intercepts of the graph of the function. The x-intercept(s) islare. Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.) The y-intercept isn. Simplify your answer. Type an integer or a fraction.) c) Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each x-intercept. The zero(s) of f islaren. Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.) The least zero is a zero of multiplicity. so the graph of f V the x-axis at x=. The middle zero is a zero of multiplicity so the graph of f V the x-axis at x =. The greatest zero is a zero of multiplicity . so the graph of V the x-axis at x=
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.
![Analyze the polynomial function \( f(x) = (x + 4)(x - 5)(x + 6) \) using parts (a) through (e).
(a) Determine the end behavior of the graph of the function.
The graph of \( f \) behaves like \( y = \) [ ] for large values of \( |x| \).
(b) Find the x- and y-intercepts of the graph of the function.
The x-intercept(s) is/are [ ].
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)
The y-intercept is [ ].
(Simplify your answer. Type an integer or a fraction.)
(c) Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each x-intercept.
The zero(s) of \( f \) is/are [ ].
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)
The least zero is a zero of multiplicity [ ], so the graph of \( f \) [crosses/touches] the x-axis at \( x = \) [ ]. The middle zero is a zero of multiplicity [ ], so the graph of \( f \) [crosses/touches] the x-axis at \( x = \) [ ]. The greatest zero is a zero of multiplicity [ ], so the graph of \( f \) [crosses/touches] the x-axis at \( x = \) [ ].
(d) Determine the maximum number of turning points on the graph of the function.
[ ] (Type a whole number.)
(e) Use the above information to draw a complete graph of the function. Choose the correct graph below.
Diagram A:
- A graph with a cubic polynomial shape, starting from the top right, curving downward, crossing the x-axis thrice, and ending at the top left.
Diagram B:
- A graph with a cubic polynomial shape, starting from the bottom left, rising, dipping below the x-axis, and rising again to end at the top right.
Diagram C:
- A graph with a cubic polynomial shape, starting from the top left, dipping below the x-axis at three distinct points, and rising to the top right.
Diagram D:
- A graph with a cubic polynomial](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f615376-0379-4c51-9f7a-08d0b15bbb48%2Fa8a531ed-4096-4d04-96db-4eedacb86d02%2Ftvauat_processed.png&w=3840&q=75)
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