Questions: Problem 3: Sketch a graph of a function that satisfies the following conditions (note: there is no one unique answer), and be sure to label all the points: a. a continuous function with an absolute minimum of -3 at x=-2 and a local minimum of -1 at x=2. b. a continuous function with no absolute extrema but with a local minimum of -2 at x=-1 and a local maximum of 2 at x=1. c. a continuous function that is increasing on (-∞, 2) and decreasing on (2,+∞).

Solution Steps

We need to create a continuous function with the following properties:

• An absolute minimum of -3 at \( x = -2 \) and a local minimum of -1 at \( x = 2 \).

One possible function that meets these conditions is: \[ y = (x + 2)^2 - 3 \]

We need to create a continuous function with the following properties:

• No absolute extrema but with a local minimum of -2 at \( x = -1 \) and a local maximum of 2 at \( x = 1 \).

One possible function that meets these conditions is: \[ y = -x^3 + 3x \]

We need to create a continuous function with the following properties:

• Increasing on \( (-\infty, 2) \) and decreasing on \( (2, +\infty) \).

One possible function that meets these conditions is: \[ y = -x^2 + 4x \]

Final Answer