Find the critical point of the function y=-2(x-1)^2-3. Then, determine whether the point represents a maximum, a minimum, or a point of inflection.
To find the critical point of the function y = -2(x-1)^2 - 3, we need to first take the derivative of the function with respect to x.
Let's start by finding the derivative:
dy/dx = -2 * 2(x-1) * 1
Simplifying this expression:
dy/dx = -4(x-1)
Next, we need to find when the derivative is equal to zero to determine the critical point. Set dy/dx = 0 and solve for x:
-4(x-1) = 0
Dividing both sides by -4:
x - 1 = 0
Adding 1 to both sides:
x = 1
So the critical point occurs at x = 1.
Now, to determine whether this point represents a maximum, a minimum, or a point of inflection, we can use the second derivative test. We need to calculate the second derivative of the function.
Taking the second derivative:
d^2y/dx^2 = -4
Since the second derivative is a constant, and it is negative (-4), this indicates that the critical point represents a maximum.
Therefore, the critical point of the function y = -2(x-1)^2 - 3 is (1, -3), and it represents a maximum.