Home
Class 12
MATHS
Let f : [0, 1]toR be such that f(xy)=f(x...

Let `f : [0, 1]toR` be such that `f(xy)=f(x)xxf(y)` for all x, `y in [0,1]" and " f(0)ne0." if ' y=y(x)` satisfues the differential equation, `dy/dx=f(x)" with"y(0)=1," then "y(1/4)+y(3/4)` is

A

(a) 5

B

(b) 3

C

(c) 2

D

(d) 4

Text Solution

Verified by Experts

The correct Answer is:
(b)
Given , `f(xy)=f(x)cdot f(y),AAx,yin[0,1] …(i)`
Putting `x = y = 0` in Eq. (i), we get
`f(0) = f(0)cdot f(0)`
`rArr f(0)[f(0)-1]=0`
`rArr f(0)=1as f(0)ne 0`
Now, put y=0 in Eq. (i) , we get
`f(0)=f(x)cdot f(0)`
`rArr f(x)=1`
so, `dy/dx=f(x)rArr dy/dx=1`
`rArr int dy=intdx`
`rArr y=x+ C`
`therefore y(0)=1`
`therefore 1=0+C`
`rArr C=1`
`therefore y=x+1`
Now, `y(1/4)=1/4+1 =5/4 and y(3/4)=3/4+1=7/4`
`rArr y(1/4)+y(3/4)=5/4+7/4=3`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let f:[0,1] rarr R be such that f(xy)=f(x).f(y), for all x,y in [0,1] and f(0) ne 0. If y=y(x) satisfies the differential equation, dy/dx=f(x) with y(0)=1, then y(1/4)+y(3/4) is equql to

f(x+y)=f(x).f(y) for all x,yinR and f(5)=2,f'(0)=3 then f'(5) is equal to

A function f:RtoR is such that f(x+y)=f(x).f(y) for all x.y inR and f(x)ne0 for all x inR . If f'(0)=2 then f'(x) is equal to

A function y=f(x) satisfies the differential equation (d y)/(d x)+x^2 y=-2 x, f(1)=1 . The value of |f^( prime prime)(1)| is

The solution of the differential equation (xy^4 + y) dx-x dy = 0, is

Let y=f(x) be a function satisfying the differential equation (x d y)/(d x)+2 y=4 x^2 and f(1)=1 . Then f(-3) is equal to

If for a function f : R to R f (x +y ) =F(x ) + f(y) for all x and y then f(0) is

If y=mx+c " and " f(0)=f'(0)=1, " then " f(2) is

A function f: R -> R satisfy the equation f (x)f(y) - f (xy)= x+y for all x, y in R and f(y) > 0 , then