Please, I want to solve the question attached in the picture. ### Calculus Problem Set**1. Analyze the Function**Given the following function:\[ Y = -x^4 + 4x^3 + 2x^2 - 6x + 8 \]For this function, you need to:a) **Find the Points of Zero Gradient.** - Determine where the derivative of the function is equal to zero.b) **Classify the Points of Zero Gradient.** - Determine if these points are maxima or minima by using the second derivative test or analyzing the nature of the function at these points.c) **Plot the Function.** - Graph the function for $ x $ values ranging from -2 to 4. - Ensure the graph clearly marks the points where the gradient is zero and specify these as maxima or minima where applicable.

Given information: Function: .To find: (a) Points of zero gradient. (b) Determine if the points of…

Answered: 1. Given the following function, Y = -x…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please, I want to solve the question attached in the picture.

### Calculus Problem Set

**1. Analyze the Function**

Given the following function:

\[ Y = -x^4 + 4x^3 + 2x^2 - 6x + 8 \]

For this function, you need to:

a) **Find the Points of Zero Gradient.**

- Determine where the derivative of the function is equal to zero.

b) **Classify the Points of Zero Gradient.**

- Determine if these points are maxima or minima by using the second derivative test or analyzing the nature of the function at these points.

c) **Plot the Function.**

- Graph the function for $ x $ values ranging from -2 to 4.
- Ensure the graph clearly marks the points where the gradient is zero and specify these as maxima or minima where applicable.

Expert Solution

Step 1: Introduction.

Given information:
Function: $y equals negative x to the power of 4 plus 4 x cubed plus 2 x squared minus 6 x plus 8$ .

To find:
(a) Points of zero gradient.
(b) Determine if the points of zero gradient are maxima or minima.
(c) Plot the function in the $x$ -value range of $negative 2$ to $4$ , including the points of zero gradient.

Concepts used:
(a) Points of zero gradient are found where the derivative of the function is equal to,, including the points of zero gradient.

Formulas used:
(a) Derivative of $x to the power of n$ with respect to $x$ is $n x to the power of n minus 1 end exponent$ .
(b) Second derivative test: If $y apostrophe apostrophe greater than 0$ , it's a local minimum; if $y apostrophe apostrophe less than 0$ , it's a local maximum.