- Define the polynomial as \(p(x) = a(x+2)(x - 0.5)(x - 2)\) based on its roots at \(-2, 0.5, 2\).
- Calculate the derivative \(p'(x) = a(3x^2 - x - 4)\) and verify that \(x = -1\) is a critical point.
- Ensure the second derivative test \(p''(-1) < 0\) holds to confirm a relative maximum at \(x = -1\), which requires \(a > 0\).
- Identify the graph that crosses the x-axis at \(-2, 0.5, 2\) and exhibits a peak at \(x = -1\).
Explanation
- Analysis of the Polynomial Function
The polynomial function has x-intercepts at \(-2\), \(\frac{1}{2}\), and \(2\). This implies that the function can be expressed in the form \(p(x) = a(x+2)(x - \frac{1}{2})(x - 2)\) for some non-zero constant \(a\). Expanding this expression, we have:
\[p(x) = a(x+2)(x^2 - 2.5x + 1) = a(x^3 - 0.5x^2 - 4x + 2)\]
We are given that there is a relative maximum at \(x = -1\). To find the critical points, we take the derivative of \(p(x)\) with respect to \(x\):
\[p'(x) = a(3x^2 - x - 4)\]
Setting the derivative equal to zero to find the critical points:
\[3x^2 - x - 4 = 0\]
Factoring the quadratic equation:
\[(3x - 4)(x + 1) = 0\]
This confirms critical points at \(x = -1\) and \(x = \frac{4}{3}\). We are given a relative maximum at \(x = -1\). To verify it is a maximum, we look at the second derivative:
\[p''(x) = a(6x - 1)\]
Evaluating at \(x = -1\):
\[p''(-1) = a(6(-1) - 1) = -7a\]
For \(p(x)\) to have a relative maximum at \(x = -1\), the second derivative must be negative (\(p''(-1) < 0\)). This implies \(-7a < 0\), so \(a > 0\). Thus, any polynomial of this form with a positive leading coefficient \(a\) will have a relative maximum at \(x = -1\).
- Geometric Properties of the Graph
The graph of the polynomial \(p(x) = a(x+2)(x - 0.5)(x - 2)\) with \(a > 0\) must satisfy the following:
- It crosses the x-axis at \(x = -2\), \(x = 0.5\), and \(x = 2\).
- Since \(a > 0\), the function goes to \(-\infty\) as \(x \to -\infty\) and to \(+\infty\) as \(x \to +\infty\).
- It has a local maximum at \(x = -1\) and a local minimum at \(x = \frac{4}{3}\).
- Between \(x = -2\) and \(x = 0.5\), the function increases towards the local maximum at \(x = -1\) and then decreases.
- Between \(x = 0.5\) and \(x = 2\), the function decreases to a local minimum at \(x = 1.33\) and then increases for \(x > 2\).
Any graph matching these geometric properties is the correct representation.
Examples
In engineering, modeling the trajectory of a projectile that dips below and above the ground level can be described by polynomial functions. For example, if a sensor measures the altitude of a moving object relative to a reference level, the points where it crosses the zero line represent the moments it passes through that reference level. Knowing the roots and the locations of local maxima (highest point) and minima (lowest point) is crucial for predicting when the object will reach critical stages in its flight, ensuring proper control and safety measures.