Question

(1) Sketch the graph of a function on [1,2] that has an absolute maximum but no absolute minimum.

(2) Sketch the graph of a function that has three local minima, two local maxima

Answer 1

To sketch the graph of a function on [1,2] that has an absolute maximum but no absolute minimum, you can use a quadratic function as an example. Calculate the vertex of the function to find the absolute maximum and plot the points on the interval [1,2]. Connect the points with a smooth curve.

Step-by-step explanation:

To sketch a graph of a function on [1,2] that has an absolute maximum but no absolute minimum, we can use a simple example of a quadratic function. Let's consider the function f(x) = -x^2 + 3x + 2. To find the absolute maximum of this function, we need to calculate the vertex. The formula for the x-coordinate of the vertex is given by x = -b/2a, where a, b, and c are the coefficients of the quadratic function. In our case, a = -1 and b = 3, so the x-coordinate of the vertex is x = -3/(-2) = 3/2. Plugging this value back into the function, we find that f(3/2) = 25/4, which is the absolute maximum. To sketch the graph, plot some points on the interval [1,2] and connect them:

x = 1: f(1) = 4

x = 3/2: f(3/2) = 25/4 (absolute maximum)

x = 2: f(2) = 2

Join these points with a smooth curve, and the resulting graph will have an absolute maximum at x = 3/2 and no absolute minimum.