24 in. 8- 2x a. Write a function to represent the volume in terms of x. b. What value of x will maximize the volume of water that can be carried by the gutter? Analyze the graph of the given function f as follows: (a) Determine the end behavior of the graph of the function. (b) Find the x- and y-intercepts of the graph. (c) Determine whether the graph crosses or touches the x-axis at each x-intercept. (d) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points. (f) Find the domain of f. Use the graph to find the (f) Use the graph to determine where f is increasing and where f is decreasing. range of f. 39) f(x) = x2(x + 3) %3D 39) Form a polynomial f(x) with real coefficients having the given degree and zeros. 40) Degree: 4; zeros: -1, 2, and 1 - 2i. 40) Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over numbers. 41) f(x) = 4x3 - 11x2 – 6x + 9 41) List the potential rational zeros of the polynomial function. Do not find the zeros.
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.
F(x)=x2(x+3)
- Determine the end behavior of the graph of the function
- Find the x- and y- intercepts of the graph
- Determine whether the graph crosses or touches the x-axis at each x-intercept.
- Use the information obtained in (a)-(e) to draw a complete graoh of f by hand. Label all intercepts and turning points
- Find the domain of f. Use the graph to find the range of f.
- Use the graph to determine where f is increasing and where f is decreasing.
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