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!
y = ax^3+bx^2+cx
y' = 3ax^2+2bx+c
y" = 6ax+2b
Since y"=0 at x = -1, -6a+2b=0
3a-b = 0
y'=4 at x = -1, so 3a-2b+c = 4
y=-5 at x = -1, so -a+b-c = -5
Now you have the three equations
3a-b = 0
3a-2b+c = 4
a-b+c = 5
y = x^3+3x^2+7x
The tangent line at (-1,-5) has slope 4, so it is y = 4x-1
See the graph at
http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E3%2B3x%5E2%2B7x,+y%3D4x-1+where+-3+%3C%3D+x+%3C%3D+1
To determine the values of a, b, and c so that the curve y = ax^3 + bx^2 + cx will have a slope of 4 at its point of inflection (-1, -5), we need to follow a step-by-step process.
Step 1: Find the first and second derivatives of the curve:
To find the slope at the point of inflection, we need to find the first derivative of the curve, y'(x), and the second derivative, y''(x).
The first derivative, y'(x), represents the slope of the curve at any given point:
y'(x) = 3ax^2 + 2bx + c
The second derivative, y''(x), represents the rate at which the slope changes:
y''(x) = 6ax + 2b
Step 2: Use the given information:
The given point of inflection is (-1, -5), which means that the x-coordinate at the point of inflection is -1 and the y-coordinate is -5.
At the point of inflection, the slope of the curve is 4. So, we can set y'(x) = 4 and solve for the variables a, b, and c.
4 = 3a(-1)^2 + 2b(-1) + c
4 = 3a - 2b + c ...(Equation 1)
Step 3: Find the value of the second derivative at the point of inflection:
Now, we need to find the value of the second derivative, y''(-1).
Substitute x = -1 into the equation y''(x):
y''(-1) = 6a(-1) + 2b = -6a + 2b = ?
Step 4: Set the second derivative equal to zero:
At the point of inflection, the second derivative, y''(x), should be equal to zero, as it changes the slope from increasing to decreasing or vice versa.
Set y''(-1) equal to zero and solve for a and b:
-6a + 2b = 0 ...(Equation 2)
Step 5: Solve the system of equations:
Solve Equation 1 and Equation 2, which will give you the values of a, b, and c.
Equation 1: 4 = 3a - 2b + c
Equation 2: -6a + 2b = 0
You can use substitution, elimination, or any method to solve this system of equations.
After solving, you will obtain the values of a, b, and c that satisfy the conditions.