Question

(a) Sketch the graph of a function on \(( - 1,2)\) and it satisfies the given conditions that the graph should have an absolute maximum but no absolute minimum.

(b) Sketch the graph of a function on \(( - 1,2)\) and it satisfies the conditions that the graph is discontinuous but has absolute maximum but no absolute minimum.

Short answer

(a) The graph is sketched.

(b) The graph is sketched.

Step-by-step solution

Given data

The given function that has an absolute maximum but 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. Absolute maximum is the value of a point which is the maximum in the whole domain. Since the graph has absolute maximum, consider any greatest point on the given interval \(( - 1,2)\). Let the point be \((2,2)\). Absolute minimum is the value of a point which is the minimum in the whole domain. Since the graph should not have absolute minimum in the interval \(( - 1,2)\), make sure that the point allotted should not be included and it should be open. Draw the graph:

From Figure, it is observed that the absolute maximum occurs at \(x = 2\). Since the curve has hole at \(x = - 1\), the point is not included in the domain. Therefore, it has no absolute minimum at \(x = - 1\). If it has no hole at \(x = - 1\) then absolute minimum occurs at \(x = - 1\).

Sketch the graph

(b)

Let \(x\) be represented in the \(x\)-axis and the value of the function \(f(x)\) be represented in the \(y\)-axis. A function is said to be discontinuous, if its domain is an interval and it is not continuous at any point of that interval. Absolute maximum is any point of the domain at which the function attains its maximum value. So, consider the greatest point on the interval \(( - 1,2)\) as 2 attained at \((0,0)\). Absolute minimum is any point of the domain at which the function attains its minimum value. So, consider the least point on the interval \(( - 1,2)\) as \( - 1\) attained at \(( - 1,1)\). 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 graph is discontinuous as it has the break near origin. Also, found that the absolute maximum occurs at \(x = 0\) and absolute minimum occurs at \(x = 1\).