`star` Maxima and Minima

`star` Local maximum value and local minimum value

`star` Local maximum value and local minimum value

`\color{green} ✍️` Let f be a function defined on an interval `I.`

(a) Then `f` is said to have a maximum value in `I,` if there exists a point `c` in `I` such that

`color{green}{f (c) ≥ f (x)}` , for all x ∈ I.

`=>` The number `f(c)` is called the maximum value of `f` in `I` and the point `c` is called a point of maximum value of `f` in `I.`

(b) `f` is said to have a minimum value in `I,` if there exists a point `c` in `I` such that

`color{green}{f (c) ≤ f (x)}`, for all x ∈ I.

`=>` The number `f(c),` in this case, is called the minimum value of f in `I` and the point `c,` in this case, is called a point of minimum value of f in `I.`

`\color{green} ✍️` (c) `f` is said to have an extreme value in `I` if there exists a point c in `I` such that `f (c)` is either a maximum value or a minimum value of `f` in `I.`

`\color{green} ✍️` The number f (c), in this case, is called an extreme value of f in I and the point c is called an extreme point.

(a) Then `f` is said to have a maximum value in `I,` if there exists a point `c` in `I` such that

`color{green}{f (c) ≥ f (x)}` , for all x ∈ I.

`=>` The number `f(c)` is called the maximum value of `f` in `I` and the point `c` is called a point of maximum value of `f` in `I.`

(b) `f` is said to have a minimum value in `I,` if there exists a point `c` in `I` such that

`color{green}{f (c) ≤ f (x)}`, for all x ∈ I.

`=>` The number `f(c),` in this case, is called the minimum value of f in `I` and the point `c,` in this case, is called a point of minimum value of f in `I.`

`\color{green} ✍️` (c) `f` is said to have an extreme value in `I` if there exists a point c in `I` such that `f (c)` is either a maximum value or a minimum value of `f` in `I.`

`\color{green} ✍️` The number f (c), in this case, is called an extreme value of f in I and the point c is called an extreme point.

Q 3135145962

Find the maximum and the minimum values, if any, of the function `f` given by

`f (x) = x^2, x ∈ R` .

Class 12 Chapter 6 Example 26

`f (x) = x^2, x ∈ R` .

Class 12 Chapter 6 Example 26

From the graph of the given function (Fig 6.10),

we have f (x) = 0 if x = 0. Also

f (x) ≥ 0, for all x ∈ R.

Therefore, the minimum value of f is 0 and the point

of minimum value of f is x = 0. Further, it may be observed

from the graph of the function that f has no maximum

value and hence no point of maximum value of f in R.

Note : If we restrict the domain of f to `[– 2, 1]` only, then `f` will have maximum value `(– 2)^2 = 4` at `x = – 2.`

Q 3165145965

Find the maximum and minimum values

of f , if any, of the function given by f (x) = | x |, x ∈ R.

Class 12 Chapter 6 Example 27

of f , if any, of the function given by f (x) = | x |, x ∈ R.

Class 12 Chapter 6 Example 27

From the graph of the given function

(Fig 6.11) , note that

f (x) ≥ 0, for all x ∈ R and f (x) = 0 if x = 0.

Therefore, the function f has a minimum value 0

and the point of minimum value of f is x = 0. Also, the

graph clearly shows that f has no maximum value in

R and hence no point of maximum value in R.

Note : (i) If we restrict the domain of f to `[– 2, 1]` only, then f will have maximum value `|– 2| = 2.`

(ii) One may note that the function `f` in Example is not differentiable at `x = 0`

Q 3115145969

Find the maximum and the minimum values, if any, of the function

given by

f (x) = x, x ∈ (0, 1).

Class 12 Chapter 6 Example 28

given by

f (x) = x, x ∈ (0, 1).

Class 12 Chapter 6 Example 28

The given function is an increasing (strictly) function in the given interval

(0, 1). From the graph (Fig 6.12) of the function f , it

seems that, it should have the minimum value at a

point closest to 0 on its right and the maximum value

at a point closest to 1 on its left. Are such points

available? Of course, not. It is not possible to locate

such points. In fact, if a point `x_0` is closest to 0, then

we find ` (x_0)/2 < x_0` for all `x_0 ∈ (0,1) ` . Also, if `x_1` is

closest to 1, then ` (x_1 +1)/2 > x_1` for all `x_1 ∈(0,1)` .

Therefore, the given function has neither the maximum value nor the minimum

value in the interval (0,1).

`"Remark:"` The reader may observe that in Example 28, if we include the points 0 and 1 in the domain of `f` , i.e., if we extend the domain of f to `[0,1],` then the function f has minimum value 0 at `x = 0` and maximum value 1 at `x = 1.` Infact, we have the following results (The proof of these results are beyond the scope of the present text)

`=>` By a monotonic function `f` in an interval `I,` we mean that `f` is either increasing in `I` or decreasing in `I.`

`=>color{green}{"Every monotonic function assumes its maximum/minimum value at the"}`

`color{green}{" end points of the domain of definition of the function."}`

`=>` Every continuous function on a closed interval has a maximum and a minimum value.

`=>color{green}{"Every monotonic function assumes its maximum/minimum value at the"}`

`color{green}{" end points of the domain of definition of the function."}`

`=>` Every continuous function on a closed interval has a maximum and a minimum value.

`=>` Observe that at points A, B, C and D on the graph, the function changes its nature from decreasing to increasing or vice-versa. These points may be called turning points of the given function.

`=>` The function has maximum value in some neighbourhood of points `B` and `D` which are at the top of their respective hills. For this reason, the points A and C may be regarded as points of local minimum value (or relative minimum value) and points B and D may be regarded as points of local maximum value (or relative maximum value) for the function

`\color{green} ✍️` Local maxima and local minima are the largest and smallest value of the function, either within a given range (the local or relative extrema) or in an entire domain of a function .

`\color{green} ✍️` Global maxima and Global minima are the largest and smallest value of the function, either within a given range or entire domain of a function

See Fig =>

`=>` The function has maximum value in some neighbourhood of points `B` and `D` which are at the top of their respective hills. For this reason, the points A and C may be regarded as points of local minimum value (or relative minimum value) and points B and D may be regarded as points of local maximum value (or relative maximum value) for the function

`\color{green} ✍️` Local maxima and local minima are the largest and smallest value of the function, either within a given range (the local or relative extrema) or in an entire domain of a function .

`\color{green} ✍️` Global maxima and Global minima are the largest and smallest value of the function, either within a given range or entire domain of a function

See Fig =>

`\color{green} ✍️` Let `f` be a real valued function and let `c` be an interior point in the domain of `f.`

(a) `c` is called a point of local maxima if there is an `h > 0` such that

`color{red}{f (c) ≥ f (x)}`, for all x in `(c – h, c + h)`

`=>` The value f (c) is called the local maximum value of f.

(b) `=>c` is called a point of local minima if there is an `h > 0` such that

`color{red}{f (c) ≤ f (x),}` for all `x` in `(c – h, c + h)`

`=>` The value f (c) is called the local minimum value of `f .`

`color{blue}{"Note :"}` The function `f` is increasing (i.e., `(f ′(x) > 0)` in the interval `(c – h, c)` and decreasing (i.e., `(f ′(x) < 0)` in the interval `(c, c + h).`

`=>`Similarly, if `c` is a point of local minima of `f` , then the graph of `f` around `c` will be as shown in fig.

Here f is decreasing (i.e., `f ′(x) < 0)` in the interval (c – h, c) and increasing (i.e., `(f ′(x) > 0)` in the interval `(c, c + h).`

(a) `c` is called a point of local maxima if there is an `h > 0` such that

`color{red}{f (c) ≥ f (x)}`, for all x in `(c – h, c + h)`

`=>` The value f (c) is called the local maximum value of f.

(b) `=>c` is called a point of local minima if there is an `h > 0` such that

`color{red}{f (c) ≤ f (x),}` for all `x` in `(c – h, c + h)`

`=>` The value f (c) is called the local minimum value of `f .`

`color{blue}{"Note :"}` The function `f` is increasing (i.e., `(f ′(x) > 0)` in the interval `(c – h, c)` and decreasing (i.e., `(f ′(x) < 0)` in the interval `(c, c + h).`

`=>`Similarly, if `c` is a point of local minima of `f` , then the graph of `f` around `c` will be as shown in fig.

Here f is decreasing (i.e., `f ′(x) < 0)` in the interval (c – h, c) and increasing (i.e., `(f ′(x) > 0)` in the interval `(c, c + h).`

`\color{green} ✍️` Let `f` be a function defined on an open interval `I.` Suppose `c ∈ I` be any point.

If `f` has a local maxima or a local minima at `x = c,` then either `f ′(c) = 0` or `color{red}{f \ \ "is not differentiable at" \ \ c.}`

`color{blue}{"Remark-"}` The converse of above theorem need not be true, that is, a point at which the derivative vanishes need not be a point of local maxima or local minima.

`=>` For example, if `f (x) = x^3`, then `f′(x)= 3x^2` and so `f′(0) = 0.` But `0` is neither a point of local maxima nor a point of local minima .

If `f` has a local maxima or a local minima at `x = c,` then either `f ′(c) = 0` or `color{red}{f \ \ "is not differentiable at" \ \ c.}`

`color{blue}{"Remark-"}` The converse of above theorem need not be true, that is, a point at which the derivative vanishes need not be a point of local maxima or local minima.

`=>` For example, if `f (x) = x^3`, then `f′(x)= 3x^2` and so `f′(0) = 0.` But `0` is neither a point of local maxima nor a point of local minima .