The Curve Y = x^4 + px + q Has a Point of Inflection (5/6), 19/36), Where p and q Are Constants. (a) Find
Understanding the properties of polynomial functions is fundamental in calculus and algebra. In particular, analyzing the behavior of quartic functions such as y = x^4 + px + q provides insights into their critical points, inflection points, and how the coefficients p and q influence the overall shape of the curve. Given that the curve y = x^4 + px + q has a point of inflection at (5/6, 19/36), our goal is to determine the values of the constants p and q. This involves leveraging calculus concepts such as derivatives and the conditions for inflection points.
In this article, we will explore the step-by-step process to find p and q, starting with the key concepts involved, then formulating the necessary equations, and finally solving them to obtain the desired constants. Whether you're a student preparing for exams or an enthusiast interested in polynomial behavior, understanding this problem deepens your grasp of calculus applications.
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Understanding Inflection Points in Polynomial Functions
What is an Inflection Point?
An inflection point on a curve is a point where the curve changes concavity. That is, the second derivative of the function changes sign at that point. This is different from a maximum or minimum, which are critical points where the first derivative is zero, but the concavity remains the same.Conditions for an Inflection Point
For a function y = f(x), a point x = a is an inflection point if:- f''(a) = 0 or f''(a) is undefined
- The second derivative changes sign at x = a
In our case, the function is y = x^4 + px + q, which is smooth and differentiable everywhere, so the key condition simplifies to setting the second derivative to zero at the inflection point.
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Given Data and Objectives
The problem states:
- The curve y = x^4 + px + q has a point of inflection at (5/6, 19/36).
- p and q are constants to be determined.
Our goals:
- Find the values of p and q that satisfy these conditions.
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Step 1: Derivatives of the Function
To find p and q, we need to analyze the derivatives:
First Derivative (dy/dx)
Given y = x^4 + px + q, \[ dy/dx = 4x^3 + p \]Second Derivative (d²y/dx²)
Differentiating again: \[ d^2y/dx^2 = 12x^2 \]Note that the second derivative depends only on x, which simplifies the condition for inflection points.
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Step 2: Applying Inflection Point Conditions
Since (5/6, 19/36) is an inflection point:
- The second derivative at x = 5/6 must be zero:
d^2y/dx^2 \bigg|_{x=5/6} = 0
\]
- The point must lie on the curve:
y(5/6) = 19/36
\]
Analyzing the Second Derivative
\[ d^2y/dx^2 = 12x^2 \] At x = 5/6: \[ 12 \times \left(\frac{5}{6}\right)^2 = 12 \times \frac{25}{36} = 12 \times \frac{25}{36} = \frac{12 \times 25}{36} = \frac{300}{36} = \frac{25}{3} \]Since this is not zero, the second derivative is positive at x = 5/6, indicating the curve is concave upward there. However, for a true inflection point, the second derivative must be zero or change sign.
But wait, the second derivative of the quartic y = x^4 + px + q is 12x^2, which is always non-negative and only zero at x = 0. Since the problem states the inflection point is at x = 5/6, we need to clarify the nature of the inflection point: in polynomial functions, a change in concavity occurs only at points where the second derivative changes sign, which for y = x^4 + px + q only occurs at x=0, because 12x^2 ≥ 0 everywhere.
Important Note:
The second derivative is always positive except at x=0, where it is zero. So the point at x=5/6, where the second derivative is positive, cannot be an inflection point unless the problem refers to a different context or the function is more complex.
Re-examining the problem, it's likely that the problem involves the third derivative or that the inflection point is at x=5/6 with y=19/36, and the change of concavity occurs at some other point, or perhaps the problem is slightly different.
However, given the problem as stated, the second derivative is 12x^2, which is always non-negative. So, the only potential inflection point is at x=0, where second derivative=0, but the problem specifies the inflection point at x=5/6.
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Step 3: Correct Approach Based on the Given Data
Given this apparent inconsistency, the most logical interpretation is:
- The point (5/6, 19/36) lies on the curve.
- The inflection point occurs at (5/6, 19/36), meaning the second derivative at x=5/6 is zero or the second derivative changes sign at this x-value.
Since the second derivative is 12x^2, which is positive everywhere except at x=0, it only equals zero at x=0. Therefore, the inflection point at x=5/6 suggests a different approach: perhaps the problem involves the third derivative or the conditions are different.
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Step 4: Alternative Analysis — Using the First and Second Derivatives
Given the nature of the problem, a more suitable approach is:
- Use the fact that at the inflection point (x=5/6):
- The point lies on the curve: y = x^4 + px + q
- The second derivative is zero or change sign.
But since d²y/dx² = 12x^2, which is positive at x=5/6, the second derivative is not zero, but the convexity does not change unless the third derivative also plays a role.
Therefore, the key is to check the third derivative:
\[
\frac{d^3 y}{dx^3} = 24x
\]
At x=5/6:
\[
24 \times \frac{5}{6} = 20 \neq 0
\]
Thus, the third derivative is non-zero, indicating the concavity does not change at x=5/6, which conflicts with the standard definition of an inflection point.
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Step 5: Final Resolution — Likely the Intended Problem
Given the initial confusion, the most logical interpretation is:
- The polynomial is y = x^4 + px + q
- The inflection point is at (5/6, 19/36)
- The second derivative at this point is zero, indicating a point of inflection.
Since d²y/dx² = 12x^2, setting this equal to zero gives x=0, not x=5/6, which conflicts with the provided point.
Therefore, perhaps the problem assumes the function y = x^4 + px + q has an inflection point at x=5/6, and the second derivative at that point is zero. But as per the derivatives, this cannot happen unless the function is different.
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Conclusion: Clarification and Step-by-Step Solution
Despite the potential inconsistencies, the core approach to solving for p and q remains:
- Use the point (5/6, 19/36) to set up the equation y = x^4 + px + q.
- Use the conditions for an inflection point, which involve derivatives.
Assuming the problem intends for us to find p and q given that (5/6, 19/36) lies on the curve, and the inflection point is at this x-coordinate, then:
- The point's coordinates satisfy the function:
19/36 = \left(\frac{5}{6}\right)^4 + p \times \frac{5}{6} + q
\]
Calculating:
\[
\left(\frac{5}{6}\right)^4 = \left(\frac{5}{6}\right)^2 \times \left(\frac{5}{6}\right)^2 = \frac{25}{36} \times \frac{25}{36} = \frac{625}{1296}
\]
So,
\[
19/36 = \frac{625}{1296} + \frac{5
