`\color{green} ✍️` Let `f` be continuous on `[a, b]` and differentiable
on the open interval (a,b). Then
`1. color{blue}{ f \ \ "is increasing in" [a,b] "if" \ \ f′(x) > 0 \ \ "for each"\ \ x ∈ (a, b)}`
`2. color{blue}{ f \ \ "is decreasing in" \ \ [a,b] \ \ "if" \ \ f ′(x) < 0 \ \ "for each" \ \ x ∈ (a, b)}`
`3. color{blue}{ f \ \ "is a constant function in" \ \[a,b] "if" \ \ f ′(x) = 0 \ \ "for each" \ \ x ∈ (a, b)}`
`color{red}{"Proof: "}` (a) Let `x_1, x_2 ∈ [a, b]` be such that `x_1 < x_2`.
Then, by Mean Value Theorem , there exists a point c between `x_1` and `x_2` such that
`f (x_2) – f (x_1) = f ′(c) (x_2 – x_1)`
i.e. `f (x_2) – f (x_1) > 0` (as f ′(c) > 0 (given))
i.e. `f (x_2) > f (x_1)`
Thus, we have
`x_1 < x_2 ⇒ f (x_1) < f (x_2 )`, for all `x_1, x_2 ∈ [a,b]`
Hence, f is an increasing function in [a,b].
The proofs of part (b) and (c) are similar manner.
`color{green} {✍️ "Key Points"}`
(i) `f` is strictly increasing in `(a, b)` if `f ′(x) > 0` for each `x ∈ (a, b)`.
(ii) `f` is strictly decreasing in `(a, b)` if `f ′(x) < 0` for each `x ∈ (a, b)`.
(iii) A function will be increasing (decreasing) in `R` if it is so in every interval of `R`.
`\color{green} ✍️` Let `f` be continuous on `[a, b]` and differentiable
on the open interval (a,b). Then
`1. color{blue}{ f \ \ "is increasing in" [a,b] "if" \ \ f′(x) > 0 \ \ "for each"\ \ x ∈ (a, b)}`
`2. color{blue}{ f \ \ "is decreasing in" \ \ [a,b] \ \ "if" \ \ f ′(x) < 0 \ \ "for each" \ \ x ∈ (a, b)}`
`3. color{blue}{ f \ \ "is a constant function in" \ \[a,b] "if" \ \ f ′(x) = 0 \ \ "for each" \ \ x ∈ (a, b)}`
`color{red}{"Proof: "}` (a) Let `x_1, x_2 ∈ [a, b]` be such that `x_1 < x_2`.
Then, by Mean Value Theorem , there exists a point c between `x_1` and `x_2` such that
`f (x_2) – f (x_1) = f ′(c) (x_2 – x_1)`
i.e. `f (x_2) – f (x_1) > 0` (as f ′(c) > 0 (given))
i.e. `f (x_2) > f (x_1)`
Thus, we have
`x_1 < x_2 ⇒ f (x_1) < f (x_2 )`, for all `x_1, x_2 ∈ [a,b]`
Hence, f is an increasing function in [a,b].
The proofs of part (b) and (c) are similar manner.
`color{green} {✍️ "Key Points"}`
(i) `f` is strictly increasing in `(a, b)` if `f ′(x) > 0` for each `x ∈ (a, b)`.
(ii) `f` is strictly decreasing in `(a, b)` if `f ′(x) < 0` for each `x ∈ (a, b)`.
(iii) A function will be increasing (decreasing) in `R` if it is so in every interval of `R`.