Determine the end behavior of the following function as x → -o: 49x2 – 3 - r(x): -9x 3 -
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.
![**Determine the end behavior of the following function as \( x \to -\infty \):**
\[
r(x) = \frac{\sqrt{49x^2 - 3}}{-9x - 3}
\]
This expression represents a rational function where the numerator contains a square root of a quadratic term, and the denominator is a linear expression. To analyze the end behavior of \( r(x) \) as \( x \) approaches negative infinity, it is important to consider the leading terms of both the numerator and the denominator.
For the numerator, \( \sqrt{49x^2 - 3} \), the dominant term as \( x \) becomes very large (in magnitude) is \( \sqrt{49x^2} = 7|x| \). Since \( x \) approaches negative infinity, this simplifies to \( 7(-x) \).
For the denominator, \(-9x - 3\), the dominant term is \(-9x\).
Thus, the function simplifies to:
\[
r(x) \approx \frac{7(-x)}{-9x} = \frac{-7x}{-9x} = \frac{7}{9}
\]
Therefore, the end behavior of the function as \( x \to -\infty \) is that \( r(x) \) approaches \(\frac{7}{9}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04f81413-92bd-4603-a7a7-bd3e00e50a8a%2Fd1cb14df-b82b-4964-b624-1cc5282ef28d%2F59fpmus5k_processed.png&w=3840&q=75)
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