Monotonic function `f` in an interval `I` means `f` is either is either increasing or decreasing.
To check increasing, decreasing , strickly Increasing, strickly decreasing find `f'(x)`
(i) `f` is strickly increasing in `(a,b)` if `f'(x) > 0` for each `x in (a,b)`
(ii) `f` is strickly decreasing in `(a, b)` if `f ′(x) < 0` for each `x ∈ (a, b)`
1. Let f be continuous on `[a, b]` and differentiable on the open interval `(a,b)`. Then
(a) `f` is increasing in `[a,b]` if `f ′(x) > 0` for each `x ∈ (a, b)`
(b) `f` is decreasing in `[a,b]` if `f ′(x) < 0` for each `x ∈ (a, b)`
(c) `f` is a constant function in `[a,b]` if `f ′(x) = 0` for each `x ∈ (a, b)`
(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. It is left as an exercise to the reader.
Monotonic function `f` in an interval `I` means `f` is either is either increasing or decreasing.
To check increasing, decreasing , strickly Increasing, strickly decreasing find `f'(x)`
(i) `f` is strickly increasing in `(a,b)` if `f'(x) > 0` for each `x in (a,b)`
(ii) `f` is strickly decreasing in `(a, b)` if `f ′(x) < 0` for each `x ∈ (a, b)`
1. Let f be continuous on `[a, b]` and differentiable on the open interval `(a,b)`. Then
(a) `f` is increasing in `[a,b]` if `f ′(x) > 0` for each `x ∈ (a, b)`
(b) `f` is decreasing in `[a,b]` if `f ′(x) < 0` for each `x ∈ (a, b)`
(c) `f` is a constant function in `[a,b]` if `f ′(x) = 0` for each `x ∈ (a, b)`
(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. It is left as an exercise to the reader.