Determining Coefficients a, b, and c for a Cubic Function Given Specific Conditions

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Introduction: In the context of cubic functions, we often encounter scenarios where specific coefficients a, b, and c need to be determined based on given conditions. This article will guide you through the process of finding these coefficients for the cubic function y ax3 bx2 cx under the conditions that it passes through the point (-1, -5) and has a slope of 4 at its point of inflection.

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Given Conditions and Equations

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Let's start by analyzing the conditions given.

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1. The Point (-1, -5) Lies on the Curve

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The point (-1, -5) lies on the curve, which means that when x -1, y -5. This can be expressed as:

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-5 -a b - c

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This simplifies to:

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a - b c 5 ... Equation 1

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2. Slope at Point (-1, -5) is 4

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The slope of the curve at a specific point is given by the derivative of the function. Here, the slope is 4 at x -1. This can be written as:

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dy/dx 3ax2 2bx c

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At x -1, the slope is 4, so we have:

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3a - 2b c 4 ... Equation 2

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3. Point of Inflection and Second Derivative

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At the point of inflection, the second derivative of the function is zero. This is a necessary condition, and we can express it as:

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d2y/dx2 6ax 2b

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At x -1, this gives us:

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6a(-1) 2b 0

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This simplifies to:

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-6a 2b 0 ... Equation 3

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Solving the Equations

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Now, let's solve these equations step by step.

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Step 1: From Equation 3, Express b in Terms of a

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From Equation 3, we can solve for b in terms of a:

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-6a 2b 0

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Rewrite it as:

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b 3a ... Equation 4

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Step 2: Substitute b 3a in Equation 1 and Equation 2

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Substitute b 3a in Equation 1:

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a - 3a c 5

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This simplifies to:

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c 2a 5 ... Equation 5

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Substitute b 3a in Equation 2:

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3a - 2(3a) c 4

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This simplifies to:

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3a - 6a c 4

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Or:

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-3a c 4 ... Equation 6

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Now, substitute Equation 5 into Equation 6:

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-3a (2a 5) 4

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This simplifies to:

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-a 5 4

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Solving for a:

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-a -1

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Therefore:

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a 1

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Now substitute a 1 back into b 3a and c 2a 5:

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b 3(1) 3

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c 2(1) 5 7

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The coefficients are:

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a 1, b 3, c 7

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Step 3: Verify the Solution

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Let's verify the solution with the given conditions.

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1. When x -1:

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y 1(-1)3 3(-1)2 7(-1) -1 3 - 7 -5 (correct)

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2. The slope at x -1:

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dy/dx 3(1)(-1)2 2(3)(-1) 7 3 - 6 7 4 (correct)

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Conclusion

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Thus, we have determined that the coefficients for the cubic function y ax3 bx2 cx given the specific conditions are a 1, b 3, and c 7. The function can be rewritten as:

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y x3 3x2 7x

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This function satisfies all the given conditions.