how do you find the derivative of this (-x^2)/ (2) +2 and does it have any x intercepts, asypmtotes, or holes.
To find the derivative of the given function and determine its x-intercepts, asymptotes, or holes, we can follow these steps:
Step 1: Find the derivative:
To find the derivative of the given function, we can apply the power rule and constant rule.
The given function is: f(x) = (-x^2)/2 + 2
Applying the power rule, the derivative of -x^2/2 is (-2/2) * x^(2-1), which simplifies to -x.
Applying the constant rule, the derivative of 2 is 0, since the derivative of any constant is always 0.
Therefore, the derivative of the function f(x) = (-x^2)/2 + 2 is f'(x) = -x.
Step 2: Determine x-intercepts:
To find the x-intercepts of a function, we set the function equal to zero (f(x) = 0) and solve for x.
Setting -x = 0, we find that x = 0. Hence, the function has one x-intercept at x = 0.
Step 3: Determine asymptotes:
To determine asymptotes, we need to consider the limit of the function as x approaches positive or negative infinity.
As the function does not involve any rational expressions or trigonometric functions, we can conclude that it does not have any vertical, horizontal, or slant asymptotes.
Step 4: Determine holes:
To determine holes in the function, we need to look for any values of x that make the function undefined.
In this case, the function does not contain any rational expressions with factors that could result in division by zero. Therefore, it does not have any holes.
In summary, the derivative of the function f(x) = (-x^2)/2 + 2 is f'(x) = -x. The function has one x-intercept at x = 0. It does not have any asymptotes or holes.