6 2 Let f (x) = X -4x + 2 with x? ®. Find the a [f"(x)] at x-1. You will need to use a calculator graphing for pert of this problem. Round your answer to 5 deumal places.
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.
![**Mathematical Analysis of Inverse Functions**
In this exercise, you are given the function \( f(x) = x^6 - 4x^2 + 2 \) with the restriction \( x \geq 0 \).
**Problem Statement:**
Find the derivative of the inverse of \( f(x) \) (denoted as \( f^{-1}(x) \)) at \( x = 1 \). This can be mathematically expressed as:
\[ \frac{d}{dx} \left[ f^{-1}(x) \right] \text{ at } x = 1 \]
**Instructions:**
1. You will need a graphing calculator to complete part of this problem.
2. Round your final answer to 5 decimal places.
To solve this problem, you may follow these steps:
1. **Functional and Inverse Relationship:**
- Recall the relationship \( f(f^{-1}(x)) = x \), which implies \( f^{-1}(f(x)) = x \).
2. **Derivative of the Inverse Function:**
- The formula for the derivative of the inverse function is given by:
\[ \left( f^{-1} \right)'(y) = \frac{1}{f'(f^{-1}(y))} \]
3. **Specific Calculation at \( x = 1 \):**
- Determine \( f(y) = 1 \), find the corresponding \( y \) using the graphing calculator.
- Compute the derivative \( f'(x) \) of the given function \( f(x) \).
- Use the value of \( y \) found from the previous step to get \( f'(y) \).
4. **Evaluate the Expression:**
- Finally, evaluate \( \frac{1}{f'(y)} \) and round your answer to five decimal places.
Through careful steps and the use of a graphing calculator, you will be able to find the required derivative of the inverse function at the specified value.
**Note:** Precision is crucial, and rounding to the fifth decimal place ensures accuracy in mathematical calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e9d947d-304f-415e-95bf-140f5d3b17e4%2F6b7ad238-204e-48e9-a205-954250f25985%2Fg471bc2_processed.png&w=3840&q=75)
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