Final answer:
To create a sign diagram for the first and second derivatives of the function f(x) = x^(5/7), you need to find where the derivatives are zero or undefined. From the sign diagrams, you can sketch the graph and determine the location of the relative extrema at x = 0.
Step-by-step explanation:
To create a sign diagram for the first derivative, we need to find where the first derivative is equal to zero or undefined. The first derivative of f(x) = x^(5/7) is f'(x) = (5/7)x^(2/7).
To find where f'(x) = 0 or undefined, we set the numerator equal to zero and solve for x. 5x^(2/7) = 0 gives us x = 0. Since the first derivative is never undefined, we only have one critical point at x = 0. We can use this point to create our sign diagram.
For the second derivative, we need to find where the second derivative is equal to zero or undefined. The second derivative of f(x) is f''(x) = (2/7)(5/7)x^(-5/7-1). Simplifying this expression gives us f''(x) = (10/49)x^(-12/7).
Since the exponent of x is negative, the expression will be undefined when x = 0, as we cannot divide by zero. So, we have one critical point at x = 0 for the second derivative. We can now create a sign diagram for the second derivative.
Now, to sketch the graph and show the relative extrema, we use the critical points from the sign diagrams. For x < 0, f'(x) is negative, so the slope is decreasing towards zero. For x > 0, f'(x) is positive, so the slope is increasing towards zero. Using this information, we can sketch the graph and label the relative extrema at x = 0 as a minimum.