Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.


ƒ' (x) = 0 for x = 1 and 2; ƒ has an absolute maximum at x = 4; ƒ has an absolute minimum at x= 0; and ƒ has a local minimum at x = 2.

Verified step by step guidance

1. Identify the critical points where the derivative ƒ'(x) = 0, which are x = 1 and x = 2. These points are where the function could have local maxima, minima, or points of inflection.
2. Since ƒ has an absolute minimum at x = 0, the function starts at its lowest point at x = 0. This means ƒ(0) is the smallest value of the function on the interval [0, 4].
3. At x = 1, where ƒ'(x) = 0, the function could have a local extremum. However, since no specific behavior is given at x = 1, it could be a point of inflection or a local maximum/minimum. Consider the behavior of the function around this point.
4. At x = 2, the function has a local minimum, and since ƒ'(x) = 0 here, the graph should show a dip at this point. This means the function decreases to x = 2 and then increases after x = 2.
5. The function has an absolute maximum at x = 4, meaning ƒ(4) is the highest value on the interval [0, 4]. The graph should rise to this point, ensuring that no other point on the graph exceeds the value at x = 4.

Key Concepts

Derivative and Critical Points

The derivative of a function, denoted as ƒ'(x), represents the rate of change of the function at a given point. Critical points occur where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. In this question, ƒ'(x) = 0 at x = 1 and 2 suggests these are critical points where the function may change from increasing to decreasing or vice versa.

Absolute and Local Extrema

Absolute extrema refer to the highest and lowest values of a function over a specified interval, while local extrema are the highest or lowest points within a neighborhood of a point. The problem states that ƒ has an absolute maximum at x = 4 and an absolute minimum at x = 0, indicating these points are the overall highest and lowest values of the function on the interval [0, 4].

Continuity of Functions

A function is continuous if there are no breaks, jumps, or holes in its graph over a given interval. In this case, the function ƒ is specified to be continuous on [0, 4], which means it can be drawn without lifting the pencil. This property is essential for ensuring that the function behaves predictably at the endpoints and throughout the interval, particularly when identifying extrema.