Analyze the polynomial function f left parenthesis x right parenthesis equals x cubed plus 0.3 x squared minus 1.4988 x minus 0.27285.

We are given the polynomial function:

\[
f(x) = x^3 + 0.3x^2 - 1.4988x - 0.27285
\]

To analyze this polynomial, we will cover the following key points:

### 1. **Degree of the Polynomial**
The degree of the polynomial is 3, as the highest power of \(x\) is \(x^3\). Therefore, this is a cubic function. A cubic function can have up to 3 real roots and up to 2 turning points.

### 2. **End Behavior**
Since the leading term is \(x^3\), which is a positive cubic term, the end behavior of the function will be:
- As \(x \to \infty\), \(f(x) \to \infty\)
- As \(x \to -\infty\), \(f(x) \to -\infty\)

This behavior is typical for polynomials with a positive leading coefficient in cubic functions.

### 3. **Critical Points (Derivatives)**
To find the critical points, we take the derivative of the function \(f(x)\) and set it equal to 0:

\[
f'(x) = 3x^2 + 0.6x - 1.4988
\]

Setting \(f'(x) = 0\) will allow us to find the critical points, which correspond to local maxima, minima, or points of inflection.

### 4. **Inflection Point**
The inflection point occurs where the second derivative of the function equals 0. We take the second derivative \(f''(x)\):

\[
f''(x) = 6x + 0.6
\]

Setting \(f''(x) = 0\) will give us the x-coordinate of the inflection point. Solving this:

\[
6x + 0.6 = 0 \implies x = -0.1
\]

Thus, the inflection point occurs at \(x = -0.1\).

### 5. **Possible Roots**
To find the real roots (or zeros) of the polynomial, we solve:

\[
x^3 + 0.3x^2 - 1.4988x - 0.27285 = 0
\]

Finding exact roots of a cubic polynomial may require numerical methods such as the Newton-Raphson method or synthetic division.

Would you like me to compute the exact critical points and roots using numerical methods?

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Here are 5 related questions to deepen your understanding:

1. What do the critical points represent in a cubic function?
2. How do you interpret the end behavior of different degrees of polynomials?
3. Can a cubic function have exactly one real root? Why or why not?
4. What is the significance of the inflection point in the context of a cubic function?
5. How do you use derivatives to determine whether a critical point is a maximum or minimum?

**Tip:** The derivative of a function not only tells you about the slope of the function at a point but also helps identify key features like maxima, minima, and inflection points.

Analyze the polynomial function f(x) = x^3 + 0.3x^2 - 1.4988x - 0.27285.

Explore a detailed analysis of the cubic polynomial function f(x) = x^3 + 0.3x^2 - 1.4988x - 0.27285. This solution covers critical points, inflection points, end behavior, and derivative calculations. Ideal for students in Grades 11-12 or early college.

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