Inflection point of 3y=x^3+3x^2-9x+3
take the 2nd derivative, set it equal to zero and solve for x
that will be the x value of the inflection point, sub it back into the original equation to get the y
your second derivative will be a linear expression, thus easy to solve for x
Thank you for this answer. Great help. ❤️
To find the inflection point of the function 3y = x^3 + 3x^2 - 9x + 3, we need to follow these steps:
Step 1: Differentiate the function.
We will differentiate the function with respect to x to find its first derivative.
dy/dx = (d/dx) [x^3 + 3x^2 - 9x + 3]
Using the power rule, the derivative of x^n is nx^(n-1), we get:
dy/dx = 3x^2 + 6x - 9
Step 2: Find the second derivative.
We will differentiate the first derivative with respect to x to find the function's second derivative.
d^2y/dx^2 = (d/dx) [3x^2 + 6x - 9]
Using the power rule again, we have:
d^2y/dx^2 = 6x + 6
Step 3: Find the x-values where the second derivative is equal to zero.
We set the second derivative equal to zero and solve for x:
6x + 6 = 0
Subtracting 6 from both sides, we get:
6x = -6
Dividing both sides by 6, we find:
x = -1
Step 4: Determine the corresponding y-value at the inflection point.
To find the corresponding y-value, substitute the x-value (-1) into the original function:
3y = (-1)^3 + 3(-1)^2 - 9(-1) + 3
3y = -1 + 3 + 9 + 3
3y = 14
y = 14/3
Step 5: Identify the inflection point.
The inflection point is the coordinate where the function changes concavity. In this case, the inflection point is (-1, 14/3).