On this question I already have f'(x) and I solved for f''(x) and its zeros. I am looking for where the graph of f(x)= (x+4) / (sqrt((x^2) + 4)) is increasing the fastest. I found the maximum to be (12-sqrt(320)) / 22 or (12-8*sqrt(5)) / 22, approximately -.26, which is incorr
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.
On this question I already have f'(x) and I solved for f''(x) and its zeros. I am looking for where the graph of f(x)= (x+4) / (sqrt((x^2) + 4)) is increasing the fastest. I found the maximum to be (12-sqrt(320)) / 22 or (12-8*sqrt(5)) / 22, approximately -.26, which is incorrect according to my online homework. Can you see where I'm going wrong? Thank you.
![The handwritten mathematical notes detail the differentiation process, likely finding critical points and analyzing maxima or minima for a given function. Here is a transcription and explanation:
1. Given Function:
\[
f(x) = (4x + 4)(x^2 + 4)^{-3/2}
\]
2. Derivative Calculation:
\[
f'(x) = (x^2 + 4)^{-3/2} \left[ (4) + (4x + 4)\left(\frac{3}{2}(x^2 + 4)^{-5/2}(2x)\right) \right]
\]
3. Simplified Derivative:
\[
= (x^2 + 4)^{-5/2} \left[ -1(x^2 + 4) + 4x + 4(3x) \right]
\]
\[
= (x^2 + 4)^{-5/2} \left[ -1x^2 + 12x - 4 \right]
\]
4. Setting \( f'(x) = 0 \):
\[
\Rightarrow \, -1x^2 + 12x - 4 = 0
\]
5. Solving the Quadratic Equation:
\[
x = \frac{12 \pm \sqrt{(12)^2 - 4(1)(-4)}}{2(1)} = \frac{12 \pm \sqrt{320}}{2}
\]
Approximating the roots:
\[
\approx \frac{12 \pm 17.888}{2}
\]
Calculation leads to roots:
\[
x \approx 3.26
\]
6. Maximum/Minimum Analysis:
\[
x = \frac{12 - 8\sqrt{5}}{22} \rightarrow \text{ thorough simplification shown}
\]
The mathematics above details the derivative and critical point calculation for the given function, showing detailed steps for solving the quadratic equation derived from the derivative set to zero, indicating a maximum or minimum of the function \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb01156c3-4f42-4f24-8d9e-b5123e3d8863%2F0b665d56-6671-4f3b-8067-c66ae9328394%2F9vhcews_processed.jpeg&w=3840&q=75)
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