Questions: Assessment, Composing and ing Functions Question 4 of 7 This test: 7 point(s) possible This question: 1 point(s) possible Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the polynomial function. f(x)=x^3-1.5x+3 The zeros of the function are (Round to the nearest thousandth. Use a comma to separate answers as needed.) What is the value of the relative maximum of this function? (Round to the nearest thousandth.) What is the value of the relative minimum of this function? (Round to the nearest thousandth.) What is the range of this function? A. B. C.

Solution Steps

To solve the given problem, we need to perform the following steps:

1. Find the Real Zeros: Use numerical methods to find the roots of the polynomial \( f(x) = x^3 - 1.5x + 3 \). This can be done using a root-finding algorithm like Newton's method or a library function.

2. Find Relative Maxima and Minima: Calculate the first derivative of the function to find critical points. Evaluate the second derivative at these points to determine if they are maxima or minima.

3. Determine the Range: Analyze the behavior of the polynomial as \( x \) approaches positive and negative infinity to understand the range. The range of a cubic polynomial is typically all real numbers unless restricted by specific conditions.

To find the real zeros of the polynomial function \( f(x) = x^3 - 1.5x + 3 \), we determined that the zeros are approximately: \[ \boxed{-1.784, 0.715, 0.733} \]

Next, we calculated the critical points by finding the first derivative \( f'(x) = 3x^2 - 1.5 \) and solving for \( x \). The critical points are: \[ x = -\sqrt{\frac{3}{4}} \quad \text{and} \quad x = \sqrt{\frac{3}{4}} \] Evaluating the second derivative \( f''(x) = 6x \) at these critical points, we found:

• At \( x = -\sqrt{\frac{3}{4}} \), \( f''(x) < 0 \) indicates a relative maximum.
• At \( x = \sqrt{\frac{3}{4}} \), \( f''(x) > 0 \) indicates a relative minimum.

The values of the relative maximum and minimum are: \[ \text{Relative Maximum} = 3.7071 \quad \text{and} \quad \text{Relative Minimum} = 2.2929 \] Thus, we have: \[ \boxed{3.7071} \quad \text{(relative maximum)} \] \[ \boxed{2.2929} \quad \text{(relative minimum)} \]

Since \( f(x) \) is a cubic polynomial, its range is all real numbers. Therefore, we conclude that the range of the function is: \[ \text{Range} = (-\infty, \infty) \]

Final Answer