Final answer:
Without an actual graph or function provided, I explained how to locate relative extrema by analyzing the first and second derivatives of the function. The horizonal line f(x) = 10 for 0≤x≤20, has no relative extrema. For U(x), we would look at the points where the first derivative is zero and the second derivative indicates the type of extremum.
Step-by-step explanation:
To find the locations and values of all relative extrema for the provided function, we need to analyze the graph of the function as well as its first and second derivatives. However, without the actual graph or a more specific function provided, I can only give a general approach.
To locate relative extrema, we first identify points where the first derivative (f'(x)) is zero or undefined. These points are potential locations of relative maximums or minimums. Next, we use the second derivative (f''(x)) to determine the concavity of the function around these points. If the second derivative is positive at a given point, it indicates a relative minimum; if negative, it represents a relative maximum.
For example, in the function with f(x) = 10 where 0≤x≤20, the graph of f(x) is a horizontal line at y = 10, which means there are no relative maximums or minimums within the interval (0, 20) as there is no change in the slope of the function.
In a scenario where we would have a graph such as U(x) which looks like a double potential well, the relative extrema would be found by setting the first derivative equal to zero and solving for x. If the described function is U(x), and it is negative at x = 0, then x = 0 is a relative maximum. If the second derivative is positive at x = +xQ, the function has relative minima at these points.
To label intercepts, maximums, and minimums on a graph, you would mark the points where the function crosses the x-axis (intercepts), and where the function reaches its highest and lowest points within a certain interval (maximums and minimums).