Grade 12Differential Calculus

question is …....f(x+y)=f(x)+f(y) is a odd function prove.....
i added image of solution of this question .please help me to understand solution …..i cant understand the solution which give in my book...........

Question image for question is …....f(x+y)=f(x)+f(y) is a odd functio

milind

11 Years agoGrade 12

5 Answers

Jitender Singh

Approved Tutor Answer11 Years ago

Ans:
Hello Student,
Please find answer to your question below

$f(x+y) = f(x)+f(y)$
Solution which you attached is little bit complicated. So I am gonna do this by an easier method.
By definition,
$f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$
$f(x+h) = f(x)+f(h)$
$f'(x) = \lim_{h\rightarrow 0}\frac{f(x)+f(h)-f(x)}{h}$
$f'(x) = \lim_{h\rightarrow 0}\frac{f(h)}{h}$
Since f(0) = 0, so it is zero by zero form. Apply L’hospital rule. We have,
$f'(x) = f'(0)$
$\frac{df(x)}{dx} = f'(0)$
$\int df(x) = \int f'(0)dx$
$f(x) = f'(0)x + c$
Put x = 0
$f(0) = c$
$c = 0$
$f(x) = f'(0)x$
So it is clearly odd function.

milind

11 Years ago

$f`(x) = f`(0)$ $\frac{df(x)}{dx} = f`(0)$ $\int df(x) = \int f`(0)dx$ sir i can not understand this step please explain again fully

milind

11 Years ago

$f`(x) = f`(0)$ $\frac{df(x)}{dx} = f`(0)$ $\int df(x) = \int f`(0)dx$ sir i can not understand this step please explain again fully

milind

11 Years ago

$\int df(x) = \int f`(0)dx$ sir i can not understand this step please explain again fully $\frac{df(x)}{dx} = f`(0)$ $f`(x) = f`(0)$

milind

11 Years ago

$f`(x) = f`(0)$ ….…..then $\frac{df(x)}{dx} = f`(0)$ ….…....then $\int df(x) = \int f`(0)dx$ sir i can not understand this step please explain again fully