Let f be function defined on -a-b- such that f-x- 0- for all x- a-b- Then prove that f is an increasing function on a-b-

Let us take any 2 points c_1 and c_2 such that {c_1,c_2}∈ (a,b) and c_2 = c_1+h , where h→ 0 Now, f'(c_1) =lim_h→ 0f(c_1+ ...

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Standard XII

Mathematics

Monotonically Increasing Functions

Let f be fu...

Question

Let $f$ be function defined on $[a, b]$ such that $f^{'} (x) > 0$ , for all $x \in (a, b)$ . Then prove that $f$ is an increasing function on $(a, b)$ .

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Solution

Let us take any $2$ points $c_{1}$ and $c_{2}$ such that ${c_{1}, c_{2}} \in (a, b)$ and $c_{2} = c_{1} + h$ , where $h \to 0$
Now, $f^{'} (c_{1}) = lim h \to 0 \frac{f (c_{1} + h) - f (c_{1})}{h} = \frac{f (c_{2}) - f (c_{1})}{c_{2} - c_{1}}$

Now, it is given that $f^{'} (x) > 0 \forall x \in (a, b)$ .
Therefore, $f^{'} (c_{1}) > 0$
$\Rightarrow \frac{f (c_{2}) - f (c_{1})}{c_{2} - c_{1}} > 0$
From the above fraction, we can conclude that:
1. For $c_{2} > c_{1}, f (c_{2}) > f (c_{1})$
2. For $c_{1} > c_{2}, f (c_{1}) > f (c_{2})$

Hence, $f (x)$ is an increasing function in $(a, b)$ .

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Similar questions

Let f be a function defined on [a, b] such that f'(x) > 0, for all $x ϵ$ [a, b]. Then prove that f is an increasing function on [a, b].

Q. Assertion :If

f

is differentiable on an open interval

(a, b)

such that

| f^{'} (x) | \leq M

for all

x ϵ (a, b)

, then

| f (x) - f (y) | \leq M | x - y |

for all

x, y ϵ (a, b)

. Reason: If

f (x)

is a continuous function defined on

[a, b]

such that it is differentiable on

(a, b)

, then there exists

c ϵ (a, b)

such that

f^{'} (c) = \frac{f (b) - f (a)}{b - a}

Q. Let

f

be any continuously differentiable function on

[a, b]

and twice differentiable on

(a, b)

such that

f (a) = f^{'} (a) = 0

and

f (b) = 0

. Then

Q. Let

f (x)

be a differentiable function defined on

[0, 2]

such that

f^{'} (x) = f^{'} (2 - x)

for all

x \in (0, 2),

f (0) = 1

and

f (2) = e^{2} .

Then the value of

2 \int 0 f (x) d x

Q. Let

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f (0) = 1,

f^{'} (0) = 2

and

f^{'} (x) \neq 0

for all

x \in R .

∣ ∣ ∣ \begin{matrix} f (x) & f^{'} (x) f^{'} (x) & f^{''} (x) \end{matrix} ∣ ∣ ∣ = 0,

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Let f be function defined on [a,b] such that f′(x)>0, for all x∈(a,b). Then prove that f is an increasing function on (a,b).

Let $f$ be function defined on $[a, b]$ such that $f^{'} (x) > 0$ , for all $x \in (a, b)$ . Then prove that $f$ is an increasing function on $(a, b)$ .