Is this statment true? Please explain.
At a relative extrema, f'(x) = 0
The statement is true, because f'(x) deals with the slope of a tangent line. At any extrema (maximum or minimum point), the tangent line will be horizontal, meaning that it has a slope of 0. Since f'(x) is the slope of the tangent line, we can say that f'(x)=0 at the extrema.
The statement "At a relative extrema, f'(x) = 0" is not always true. A relative extremum occurs at a point on a function where it reaches a maximum or minimum value within a certain interval.
To determine the critical points of a function, where relative extrema can potentially occur, you need to find the values of x where the derivative of the function, f'(x), equals zero or is undefined.
To explain further, here are the steps to find the critical points of a function:
1. Take the derivative of the function f(x) with respect to x to get f'(x).
2. Set f'(x) equal to zero and solve the resulting equation for x. This step helps you find the x-coordinates of points where the tangent line to the function is horizontal (horizontal tangent).
3. Check the values of x obtained from step 2 in the original function f(x) and determine if each corresponding point is a relative extremum (maximum or minimum) or an inflection point.
- If the second derivative of the function, f''(x), is positive at a point, that point is a local minimum.
- If the second derivative of the function, f''(x), is negative at a point, that point is a local maximum.
- If the second derivative of the function, f''(x), is zero or undefined at a point, additional investigation is required to determine the nature of the point.
By following these steps, you can identify the x-values where relative extrema may occur. However, it is important to note that there are cases where the derivative of the function is zero but a relative extremum does not exist. For instance, a function may have a point where the derivative is zero but there is no maximum or minimum at that point (e.g., a horizontal inflection point). Therefore, it is crucial to verify such points by analyzing the behavior of the function around them.