# Local minimum and local maximum imply that the function approaches negative and positive infinite at opposite sides of the graph.

Local minimum (1,1) and local maximum (3,3) means the slope of the function is 0 at these points.

Thank you so much.

So
1)when the local minima is (1,1) & (3,3), how can I sketch the graph?

2)when the local maxima is (1,1) & (3,3), how can I sketch the graph?

These are three different problems!

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## To sketch the graph when there are local minima at (1,1) and (3,3), follow these steps:

1) Start by plotting the points (1,1) and (3,3) on the x-y plane. These points represent the local minima.

2) Since the slope of the function is 0 at these points, draw a horizontal tangent line at each of these points. This means that the function is neither increasing nor decreasing at these points.

3) The shape of the graph between these points can vary. If you have any additional information about the behavior of the function, you can use it to guide your sketch. For example, if you know that the function is increasing before the local minimum (1,1) and after the local minimum (3,3), you can roughly sketch an increasing curve connecting these points.

4) It's important to note that between the local minima, the function can take any shape that is consistent with the given conditions. You can make the curve concave upwards or downwards to explore different possibilities.

To sketch the graph when there are local maxima at (1,1) and (3,3), follow similar steps:

1) Plot the points (1,1) and (3,3) on the x-y plane to represent the local maxima.

2) Since the slope of the function is 0 at these points, draw a horizontal tangent line at each of these points. This means that the function is neither increasing nor decreasing at these points.

3) The shape of the graph between these points can again vary. If you have additional information about the behavior of the function, you can use it to guide your sketch. For example, if you know that the function is decreasing before the local maximum (1,1) and after the local maximum (3,3), you can roughly sketch a decreasing curve connecting these points.

4) Just like in the case of local minima, between the local maxima, the function can take different shapes that align with the given conditions. You can make the curve concave upwards or downwards to explore various possibilities.

Remember that these sketches provide a rough idea of what the graph could look like based on the given information.