Determine a,b and c so that curve y= ax^3 + bx^2 + cx will have a slope of 4 at its point of inflection (-1,-5)

Help please!

-5 = -a + b - c

y' = 3a x^2 + 2b x + c
... 4 = 3a - 2b + c

y" = 6a x + 2b
... 0 = -6a + 2b

3 equations, 3 unknowns
... solve the system

You might have check my solution given earlier before reposting the problem:

http://www.jiskha.com/display.cgi?id=1486538658

To determine the values of a, b, and c, we need to use the given information about the curve's slope at the point of inflection.

1. Slope at the inflection point:
We know that the slope at the point of inflection is 4. This means that the derivative of the function at that point is equal to 4.

2. Derivative of the function:
To find the derivative of the function, y = ax^3 + bx^2 + cx, we differentiate it with respect to x.

y' = 3ax^2 + 2bx + c

3. Plug in the point of inflection:
To find the values of a, b, and c, we can substitute the given point (-1, -5) into the derivative equation.

-5 = 3a(-1)^2 + 2b(-1) + c

Simplifying the equation:

-5 = 3a - 2b + c

Now, we have a system of equations:
Equation 1: -5 = 3a - 2b + c

4. Equation for the point of inflection:
We also know that the point of inflection is (-1, -5). We can substitute this point into the original function equation.

-5 = a(-1)^3 + b(-1)^2 + c(-1)

Simplifying the equation:

-5 = -a + b - c

Equation 2: -5 = -a + b - c

Now, we have two equations:
Equation 1: -5 = 3a - 2b + c
Equation 2: -5 = -a + b - c

5. Solve the system of equations:
You can solve these equations simultaneously. Here's one approach:

First, take Equation 1 and Equation 2 and add them together:

-5 + (-5) = (3a - 2b + c) + (-a + b - c)

Simplifying the equation:

-10 = 2a - b

Now, solve this equation for b:

b = 2a + 10

Next, substitute this expression for b into Equation 1:

-5 = 3a - 2(2a + 10) + c

Simplifying the equation:

-5 = -a - 20 + c

Rearrange the equation:

a + c = 15

Now, we have two equations:

Equation 3: b = 2a + 10
Equation 4: a + c = 15

6. Solve for a, b, and c:
You can either solve these equations further by substitution or elimination method. Let's use substitution method here.

From Equation 4, solve for c:

c = 15 - a

Now, substitute this expression for c into Equation 3:

b = 2a + 10

So the values of a, b, and c are:

a = any real number
b = 2a + 10
c = 15 - a

For any value of a, you can find the corresponding values of b and c using the equations above. This will give you the values that satisfy the given conditions for the curve.

To determine the values of a, b, and c, we need to consider the properties of the curve and use a combination of differentiation and substitution.

1. The first step is to find the second derivative of the curve to determine the concavity and point of inflection. The second derivative gives us information about the slope and concavity of the curve.

Since the equation of the curve is y = ax^3 + bx^2 + cx, let's find the first derivative by differentiating with respect to x:
dy/dx = 3ax^2 + 2bx + c

The second derivative is obtained by differentiating the first derivative with respect to x:
d^2y/dx^2 = 6ax + 2b

2. Next, substitute the x-coordinate of the point of inflection (-1) into the second derivative equation and set it equal to 0, as the slope of the curve at the point of inflection is 0 (horizontal tangent).

0 = 6a(-1) + 2b
0 = -6a + 2b

3. Now, substitute the x-coordinate (-1) into the original equation y = ax^3 + bx^2 + cx and set it equal to the y-coordinate (-5) to find another equation involving a, b, and c.

-5 = a(-1)^3 + b(-1)^2 + c
-5 = -a + b + c

4. We now have two equations that we can solve simultaneously to determine the values of a, b, and c. Solve the system of equations:

-6a + 2b = 0 (Equation 1)
-a + b + c = -5 (Equation 2)

From Equation 1, we can write:
2b = 6a
b = 3a

Now substitute b = 3a into Equation 2:
-a + 3a + c = -5
2a + c = -5

Since we have two equations and two variables, we can solve for a and c:
2a + c = -5
c = -5 - 2a

5. Finally, substitute the value of c into the equation for b (b = 3a) to get the value of b in terms of a:
b = 3a

Now we have expressions for a, b, and c in terms of a. The values of a, b, and c are not unique and can take any value as long as they satisfy these relationships.

In summary:
a = any value
b = 3a
c = -5 - 2a

For example, if we let a = 1, then b = 3, and c = -7. So one possible solution is a = 1, b = 3, and c = -5, resulting in the curve y = x^3 + 3x^2 - 5x.