[Math] determine the numbers a, b, and c such that it satisfies all of the 3 conditions

calculusderivatives

I have a equation y=x^3+ ax^2 + bx + c and I need to find the numbers a, b and c such that the graph of y=x^3+ ax^2 + bx + c satisfies all of the following 3 conditions:

  1. It has a horizontal tangent line at x = -1

  2. its tangent line at x = 1 is parallel to the line y = 8x – 5.

  3. it passes through the point (1,2)

for 2), parallel means they have to have the same slope. SO derivative of the equation is 3x^2-2ax+b. When I put 1 in for x, i get 3-2a+b. When I put 1 in for y=8x-5 , I get y=3. So 3-2a+b=3, so a should be 1 b should be 2.

I'm not sure i'm doing it correctly.. Any help would be very much appeciated

Best Answer

Well,first consider (1).A horizontal tangent line means the derivative with respect to x at x=- 1 is 0. This can be seen geometrically by the fact this represents a horizontal line at y = c where c is a real constant.

Now consider (2). A line is parallel to another line when their slopes are the same, which means the derivatives are the same at x= 1. So f'(1) = 8. You're lastly given that when x = 1, y =2 on the graph of the curve of the function.

So now let's put this all together:

$$1)\frac{dy}{dx}=0 = 3x^2 + 2ax + b when x = -1 -----> 0 = 3 - 2a +b$$ $$2) \frac{dy}{dx}=8 = 3x^2 + 2ax + b when x = 1 -------> 8= 3 + 2a + b $$ $$3) At (1,2) , y= x^3 + ax^2 + bx + c -----> 2 = 1^{3}+ a*(1^{2}) +b + c = 1+a+b+c $$

You now have a series of 3 linear equations in 3 unknowns that you can solve by your favorite method. I'm a matrix guy myself.

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