Final answer:
To sketch the graph of a function that is continuous and increasing on [0, 2], has a relative minimum at x = 1, and is not differentiable at x = 1, plot a point at the relative minimum, draw an increasing curve approaching x = 1 with a sharp corner, and continue increasing smoothly after x = 1 with no cusps or corners.
Step-by-step explanation:
To sketch the graph of a function with the given descriptions, consider the following steps:
- Interval [0, 2]: The function must be continuous and increasing on this interval. This suggests a smooth curve or line that goes up as we move from left to right.
- Relative minimum at x = 1: A relative minimum is a point where the function has a lower value than at nearby points. Since the function is increasing on [0, 2], this minimum must occur at the beginning of this interval. At x = 1, the function should have the lowest point in the vicinity.
- Not differentiable at x = 1: To not be differentiable at x = 1, the graph could have a sharp corner or cusp at that point. This means that although the function is continuous, the slope of the curve changes abruptly.
With these criteria in mind, start by plotting a point for the relative minimum at x = 1. Then, draw an increasing curve from 0 to 1, making sure it approaches the point at x = 1 with a sharp corner or cusp. From x = 1 to x = 2, continue the increasing curve smoothly without any corners or cusps. Remember, the graph must stay continuous, which means that there should be no gaps or jumps in the curve.