Learn graphical & formal definition of absolute maximum and minimum values of a function. Sketch the graph of a function with desired local and absolute extrema values.

Consider the graph of a function \('f'\).

- The highest point on the graph of \('f'\) is (7, 5). We say that,

\(f(7)=5\) is the largest value the function can take.

- The lowest point on the graph of \(f\) is (4, –2). We say that,

\(f(4)=â€“2\) is the smallest value of function.

- The largest value of the function in an interval is called the absolute maximum value of the function and similarly smallest value of the function in the interval is called absolute minimum value of the function.
- 5 is the absolute maximum value of \('f'\) and –2 is the absolute minimum value of \(f\).
- height of highest point = absolute maximum of \(f\).

Height of lowest point = absolute minimum value of \(f\).

- The number \(f(c)\) is called the local maximum value of a function \('f'\) if

\(f(c)\geq f(x)\) for all \(x\) which are near \('c'\) or in some interval containing \('c'\).

\(A,\,B,\,C,\,D\) are all points of local maxima or \(f(a),\,f(b),\,f(c),\,f(d)\) are all local maximum values of function.

- The point or the value of \(x\) at which local maximum value occurs is called local maxima.
- Note that local word is used to convey that there can be more than one such maximas for the same function.

A 1, –1 and 3

B –2, 4 and 3

C \(\dfrac{1}{4}\), 2 and 6

D 4, –3 and 1

To find the absolute maximum and absolute minimum value of a function in closed interval [a, b]:

- Find all critical numbers of \(f\) in [a, b] and then value of \(f\) at these critical numbers.
- Find value of \(f\) at the end point i.e. \(f(a)\) and \(f(b)\).
- The largest of the values from step (1) and (2) is the absolute maximum and smallest of the values is absolute minimum value of the function.

A Absolute maximum value=5, Absolute minimum value =1

B Absolute maximum value= \(\dfrac{6\pi\sqrt3\,-\pi^2+18}{36}\), Absolute minimum value =1

C Absolute maximum value=1, Absolute minimum value = –1

D Absolute maximum value= \(\dfrac{6\pi+\pi^2}{3}\), Absolute minimum value =2

- Let \(c\) be \(a\) number in the domain \(D\) of \('f'\). Then \(f(c)\) is called absolute maximum or global maximum value of \(f\) in \(D\) if

\(f(c)\geq f(x)\) for all \(x\) in \(D\).

- Let \(c\) be \(a\) number in the domain \(D\) of \('f'\). Then \(f(c)\) is called absolute minimum or global minimum value of \(f\) in \(D\) if

\(f(c)\leq f(x)\) for all \(x\) in \(D\).

- The maximum and minimum values of \(f\) are together called extreme values of \(f\).

The absolute maximum value of \(f\) is \(f(\beta)\) and absolute minimum value is \(f(\alpha)\).

A The absolute maximum value of \(f\) is 5

B The absolute minimum value of \(f\) is 2

C The absolute minimum value of \(f\) is –3

D The absolute maximum value of \(f\) is 0

- The number \(f(c)\) is called the local minimum value of a function \('f'\) if

\(f(c)\leq f(x)\) for all \(x\) near \(c\) or in some interval contain \('c'\).

\(A,\,B,\,C\) are the point of local minima. \(f(a),\;f(b),\;f(c)\) are the local minimum values.

- The point or the value of x at which local minimum value occur is called local minima.
- Note that 'local' word is used to convey that there can be more than one such minima for a function.

A 0 and –2

B \(\dfrac{5}{4}\) and –1

C –1 and 2

D 1 and –2

- If some particular absolute maximum and minimum values of a function are desired then mark the highest and lowest points on the graph according to the value given.
- Suppose a function has absolute maximum at \(x=5\) and absolute minimum at \(x=-2\) then there can be many possible graphs, one of which can be.

- Similarly the intermediate top and bottom points can be added to the graph according to desired local extremas (i.e. local maximas and minimas)
- If a function \(f\) has absolute maximum at \(x=3\), absolute minimum at \(x=-4\), local maximum at \(x=1\) and local minimum at \(x=2\) then its graph can be as follows.

- If we have to find the absolute maximum and minimum value of a piecewise function i.e. a function which is defined by different expression in different intervals, we first sketch the graph of the function.
- If the compact functions of piecewise functions are linear in nature, then graph is easy to sketch.
- The highest and lowest point on the graph will give the absolute maximum and absolute minimum value of the function.

e.g. Look at the graph below. \(f\) has absolute minimum value at \(x=2\).

A Absolute maximum value = 5, Absolute minimum value =11

B Absolute maximum value = 11, Absolute minimum value = –1

C Absolute maximum value = 5, Absolute minimum value = –1

D Absolute maximum value = 2, Absolute minimum value = –3