Determine the zeros (x-intercepts) of the following function. Why can Rolle’s Theorem by applied to state that at least one critical number must lie between these intercepts? What is/are the critical number(s)? Does it/do they truly lie between the intercepts? f(x)= 3x² – 15x – 18

Q: Find the four real zeros of the function ƒ(x) = 2x4 - 4x2 + 1.

A: Given:

Q: Evaluate the following functions at the given values, and simplify the resultant expressions: a) For…

A: As per company guidelines we are supposed to answer maximum three subparts from a question. for…

Q: 21. Given the functions f(x) each: a.) (fog)(x), = 1 x + 5 and g(x) √2-x, find the following b.)…

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Q: Let f(x) = 1 - (x + 3) + 2x. Find all values of x for which f(x) is at least 4.

A: Given that, fx=1-x+3+2x Need to evaluate the values of x for which f(x) at least 4 that is fx≥4

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Q: Let F(a) be Z2. Show that a2 and a2+a are zeros of x3+x+1.

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Q: how many zeros are in the solution to the function g(x)=x2-3x+4

A: SOLUTION-

Q: Find all real and non-real roots of the function ƒ(x) = (x2 + 4)(x – 2).?

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Q: ƒ(x) = = ax² + 3x + b 9x² - 6x + 2a if x ≤ 0, if x > 0.

A: f(x)={ax2+3x+b if x≤09x2−bx+2a if x>0

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.