Suppose f and g are differentiable functions on 0- - such that f - x - g x - x and g - x - f x - X- for all x -0 - Further- f 1-3 andg1-1If f x - g x - B x m- for all x -0 and some constant B- then the value of m equalsA- 1B- 2C- 1D- -2

Let $f : (0, \infty) \to R$ be a differentiable function such that
$f^{'} (x) = 2 - \frac{f (x)}{x}$ for all $x ϵ (0, \infty)$ and $f (1) \neq 1$ . Then

Suppose f(x) and g(x) are two continuous functions defined for $0 \leq x \leq 1$ .
Given $f (x) = \int_{0}^{1} e^{x + t} . f (t) d t$ and $g (x) = \int_{0}^{1} e^{x + t} . g (t) d t + x$ .

The value of g(0)-f(0) equals

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Suppose f and g are differentiable functions on (0,∞) such that f'(x)=−g(x)x and g'(x)=−f(x)x, for all x>0. Further, f(1)=3 and g(1)=−1. If f(x)−g(x)=Bxm, for all x>0 and some constant B, then the value of m equals

The correct option is A 1ddx[f(x)−g(x)x]=x[f′(x)−g′(x)]−[f(x)−g(x)]x2=0⇒f(x)−g(x)=Bx1⇒m=1

Suppose $f$ and $g$ are differentiable functions on $(0, \infty)$ such that $f' (x) = - \frac{g (x)}{x}$ and $g' (x) = - \frac{f (x)}{x}$ , for all $x > 0$ . Further, $f (1) = 3$ and $g (1) = - 1$ .

If $f (x) - g (x) = B x^{m},$ for all $x > 0$ and some constant $B$ , then the value of $m$ equals

The correct option is A $1$
$\frac{d}{d x} [\frac{f (x) - g (x)}{x}] = \frac{x [f^{'} (x) - g^{'} (x)] - [f (x) - g (x)]}{x^{2}} = 0 \Rightarrow f (x) - g (x) = B x^{1} \Rightarrow m = 1$