Find the points on the graph of the equation y= x^4 -12x^2 which the curvature is 0.
To find the points on the graph of the equation where the curvature is 0, we need to find where the second derivative of the equation is equal to zero.
Step 1: Find the first derivative of the equation.
First, we differentiate y = x^4 - 12x^2 with respect to x.
Using the power rule, we get dy/dx = 4x^3 - 24x.
Step 2: Find the second derivative.
Next, we differentiate the first derivative (dy/dx) with respect to x.
Using the power rule again, we get d²y/dx² = 12x^2 - 24.
Step 3: Set the second derivative equal to zero and solve for x.
To find where the curvature is 0, we set the second derivative equal to zero and solve for x.
12x^2 - 24 = 0.
Adding 24 to both sides, we get 12x^2 = 24.
Dividing both sides by 12, we have x^2 = 2.
Taking the square root of both sides, we get x = ±√2.
Step 4: Determine the corresponding y-values.
To find the corresponding y-values, we substitute the x-values we found in Step 3 back into the original equation y = x^4 - 12x^2.
For x = √2, y = (√2)^4 - 12(√2)^2 = 2 - 12(2) = -22.
For x = -√2, y = (-√2)^4 - 12(-√2)^2 = 2 - 12(2) = -22.
Thus, the points on the graph where the curvature is 0 are (√2, -22) and (-√2, -22).