Question 1122638: The slope of the tangent line to the graph of a cubic polynomial is -24 at two different points: (-2,79) and (1,-20). Determine the equation of the cubic polynomial.
Found 2 solutions by josmiceli, ikleyn:
Answer by josmiceli(19441)   (Show Source):
Answer by ikleyn(52775)  (Show Source):
You can put this solution on YOUR website! .
The solution by @josmicely is TOTALLY and ABSOLUTELY wrong.
Below find the correct solution.
We are going to find the numerical values of four coefficients a, b, c and d of the polynomial of the third degree y = ax^3 + bx^2 + cx + d.
Equation 1.
I will derive this equation from the condition that the point (-2,79) lies on the graph, i.e. satisfies the equation
a*(-2)^3 + b*(-2)^2 + c*(-2) + d = 79, or
-8a + 4b -2c + d = 79. (1)
Equation 2.
I will derive this equation from the condition that the point (1,-20) lies on the graph, i.e. satisfies the equation
a*1^3 + b*1^2 + c*1 + d = -20, or
a + b + c + d = -20. (2)
Equation 3.
I will derive this equation from the condition that the slope at the point (-2,79) is -24.
Since the first derivative is y'(x) = 3ax^2 + 2bx + c, the equation for the slope is
3a*(-2)^2 + 2b*(-2) + c = -24, or
12a - 4b + c = -24. (3)
Equation 4.
I will derive this equation from the condition that the slope at the point (1,-20) is -24.
Similarly to equation 3 case, the equation for the slope is
3a*1^2 + 2b*1 + c = -24, or
3a + 2b + c = -24. (4)
So, your system of equations is
-8a + 4b -2c + d = 79 (1)
a + b + c + d = -20 (2)
12a - 4b + c = -24 (3)
3a + 2b + c = -24. (4)
Next, use some technique or technology to solve it.
Probably, your hand calculator may help.
I use an online free of charge matrix equation solver
https://matrix.reshish.com/gauss-jordanElimination.php
I recommend you to get familiar with it.
To complete your assignment, input the augmented matrix into the solver and press the "Solve" button.
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