Question

(a) Sketch the graph of a function satisfies the following conditions that the graph has two local maxima, one local minimum and no absolute minimum.

(b) Sketch the graph of a function satisfies the conditions that the graph has three local minima, two local maxima and seven critical numbers.

Short answer

(a) The graph is sketched.

(b) The graph is sketched.

Step-by-step solution

Given data

The given function that the graph has two local maxima, one local minimum and no absolute minimum.

Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

Sketch the graph

(a)

Let the \(x\) be represented in the \(x\)-axis and the value of the function \(f(x)\) be represented in the \(y\)-axis. Local maximum is the point in the interval of consideration of the domain at which the function attains its maximum value but not the highest value. Local minimum is the point in the interval of consideration of the domain at which the function attains its minimum value but not the least value. Absolute minimum is any point of the domain at which the function attains its minimum value. Draw the graph:

From Figure, it is observed that the local maxima occurs at two points at \(x = - 2\) and \(x = 0\). Local minimum occurs at \(x = - 1\). Since the graph has a hole at \(x = 1\) the point is not included in the domain. There is no absolute minimum occurs.

Sketch the graph

(b)

Let the \(x\) be represented in the \(x\)-axis and the value of the function \(f(x)\) be represented in the \(y\)-axis. Local maximum is the point in the interval of consideration of the domain at which the function attains its maximum value but not the highest value. So, consider three local maximum points on the graph. Local minimum is the point in the interval of consideration of the domain at which the function attains its minimum value but not the least value. So, take two local minimum points on the curve. Draw the graph of the function \(f\) in such a way that it satisfies the given conditions as shown below in Figure:

From Figure, it is observed that the local maximum occurs at two points such that \(x = 0\) and \(x = 2\). Local minimum occurs at three points such that \(x = - 1,x = 1\) and \(x = 3\). Also, observe that the critical numbers occur at \(x = - 2\), \(x = 4\). Critical numbers can occur on local maxima and local minima. Therefore, including local maxima and minima points there are seven critical numbers. Thus, the graph has three local minima, two local maxima and seven critical numbers.