Name: Concept of Differentiability
Uploaded: 2019-07-19
Duration: 6 min 18 s
Description: We say that 'function is differentiable'. What does this exactly mean? What is the concept behind it? Let's learn more about this in this video.

Question

If

f (x)

is a real-valued function not identically equal to zero, such that

f (x + y^{n}) = f (x) + f^{n} (y)

,

x, y \in R

and

n

is natural number

> 1

and

f' (0) \geq 0

, then find the values of

f (5)

and

f' (10)

.

Accepted Answer

Given, $f (x + y^{n}) = f (x) + f^{n} (y)$ ...(1).

Substituting $x = y = 0$ in equation (1), we get, $f (0) = 0$ .

Now, $f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \lim_{y \to 0} \frac{f (x + y^{n}) - f (x)}{y^{n}} = \lim_{y \to 0} \frac{f (x) + f^{n} (y) - f (x)}{y^{n}}$

(Put, $h = y^{n}$ , as $h \to 0 \Rightarrow y \to 0$ ).

$\Rightarrow f' (x) = \lim_{y \to 0} {(\frac{f (y)}{y})}^{n} = {(\lim_{y \to 0} \frac{f (y + 0) - f (0)}{y})}^{n} = {(f' (0))}^{n}$

$\Rightarrow f' (x) = p$ (say)

Integrating both sides, we get,

$f (x) = p x + q$ ...(2).

As, $f (0) = 0 \Rightarrow q = 0 \Rightarrow f (x) = p x$ .

Now from equation (1), replacing $f (x) = p x$ we get,

$p (x + y^{n}) = p x + {(p y)}^{n} \Rightarrow p = p^{n} \Rightarrow p = 1$ .

Therefore, $f (x) = x (∵ f (x) \neq 0)$ .

$∴ f (5) = 5 & f' (10) = 1$ .

If $f (x)$ is a real-valued function not identically equal to zero, such that $f (x + y^{n}) = f (x) + f^{n} (y)$ , $x, y \in R$ and $n$ is natural number $> 1$ and $f' (0) \geq 0$ , then find the values of $f (5)$ and $f' (10)$ .

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

If fx is a real-valued function not identically equal to zero, such that fx+yn=fx+fny, x,y∈R and n is natural number >1 and f'0≥0, then find the values of f5 and f'10.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

If $f (x)$ is a real-valued function not identically equal to zero, such that $f (x + y^{n}) = f (x) + f^{n} (y)$ , $x, y \in R$ and $n$ is natural number $> 1$ and $f' (0) \geq 0$ , then find the values of $f (5)$ and $f' (10)$ .