A function f: R rarr R" satisfies sin x ...

A function `f: R rarr R" satisfies sin x cos y "(f(2x+2y)-f(2x-2y))= cos x sin y (f(2x+2y)+f(2x-2y)).`" If "f'(0)=(1)/(2),` then

`f''(x)=f(x)=0`

`4f''(x)+f(x)=0`

`f''(x)+f(x)=0`

`4f''(x)-f(x)=0`

Text Solution

Verified by Experts

`"We have "(f(2x+2y))/(f(2x-2y))=(sin(x+y))/(sin (x-y))`
`"or "(f(alpha))/(sin""(alpha)/(2))=(f(beta))/(sin""(beta)/(2))=K`
`"or "f(x)=K sin""(x)/(2)`
`therefore" "f'(x)=(K)/(2)cos""(x)/(2)`
`"and "f''(x)=(-K)/(4)sin""(x)/(2)`
`"or "4f''(x)+f(x)=0`

Topper's Solved these Questions

DIFFERENTIATION
CENGAGE|Exercise Exercise (Multiple)|22 Videos
DIFFERENTIATION
CENGAGE|Exercise Exercise (Comprehension)|23 Videos
DIFFERENTIATION
CENGAGE|Exercise Exercise 3.9|14 Videos
DIFFERENTIAL EQUATIONS
CENGAGE|Exercise Question Bank|25 Videos
DOT PRODUCT
CENGAGE|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

A function f: R->R satisfies sinxcosy(f(2x+2y)-f(2x-2y)=cosxsiny(f(2x+2y)+f(2x-2y))dot If f^(prime)(0)=1/2,t h e n (a)f^(prime)(x)=f(x)=0 (b)4f^(x)+f(x)=0 (c)f^(x)+f(x)=0 (d)4f^(x)-f(x)=0

A function f:R rarr R^(+) satisfies f(x+y)=f(x)f(y)AA x in R If f'(0)=2 then f'(x)=

A function f:R rarr R satisfies the equation f(x+y)=f(x)f(y) for allx,y in R and f(x)!=0 for all x in R .If f(x) is differentiable at x=0 .If f(x)=2, then prove that f'(x)=2f(x) .

A function f : R rarr R satisfies the equation f(x+y) = f(x). f(y) for all x y in R, f(x) ne 0 . Suppose that the function is differentiable at x = 0 and f'(0) = 2 , then prove that f' = 2f(x) .

If a function f:R rarr R be such that f(x-f(y))=f(f(y))+xf(y)+f(x)-1AA x,y in R then f(2)=

A function f:R rarr R" satisfies the equation f(x+y)=f(x)f(y) for all values of x" and "y and for any x in R,f(x)!=0 . Suppose the function is differentiable at x=0" and "f'(0)=2 , then for all x in R,f(x)=

If the function f:R rarr R defined by f(x)=(3^(x)+3^(-x)) , then show that f(x+y) = f(x-y) = 2f(x)f(y) .

CENGAGE-DIFFERENTIATION-Exercise (Single)

If a x^2+2h x y+b y^2=1,t h e n(d^(2y))/(dx^2) is (h^2-a b)/((h x+b y)...
Text Solution
|
"If "y^(1//m)=(x+sqrt(1+x^(2)))," then "(1+x^(2))y(2)+xy(1) is (where ...
Text Solution
|
If ( sin x) (cos y) = 1//2, then d^(2)y//dx^(2) at (pi//4, pi//4) is
Text Solution
|
A function f satisfies the condition f(x)=f'(x)+f''(x)+f''(x)+…, where...
Text Solution
|
Let f(x) be a polynomial of degree 3 such that f(3)=1, f'(3)=-1, f''(3...
Text Solution
|
If y^2=a x^2+b x+c , then y^3(d^2y)/(dx^2) is (a) a constant (b) a fun...
Text Solution
|
"If "y= sin x +e^(x)," then "(d^(2)x)/(dy^(2))=
Text Solution
|
If f''(x) =- f(x) and g(x) = f'(x) and F(x)=(f((x)/(2)))^(2)+(g((x)/(2...
Text Solution
|
Let y=in (1+ cos x)^(2). The the value of (d^(2)y)/(dx^(2))+(2)/(e^(y/...
Text Solution
|
x=tcost ,y=t+sintdot Then (d^(2x))/(dy^2)a tt=pi/2 is (pi+4)/2 (b) -(p...
Text Solution
|
"Let "y=t^(10)+1 and x=t^(8)+1." Then "(d^(2)y)/(dx^(2)) is
Text Solution
|
If x = log p and y=(1)/(p), then
Text Solution
|
If x=varphi(t), y=psi(t),t h e n(d^(2y))/(dx^2) is (varphi^(prime)psi^...
Text Solution
|
If f(x)=x^4tan(x^3)-x1n(1+x^2), then the value of (d^4(f(x)))/(dx^4) ...
Text Solution
|
Let g(x) be the inverse of an invertible function f(x), which is diffe...
Text Solution
|
f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3). Then the least value of n for...
Text Solution
|
Let y=f(x) and x=(1)/(z)." If "(d^(2)y)/(dx^(2))=lamda(z^(3))(dy)/(dz)...
Text Solution
|
Let x=f(t) and y=g(t), where x and y are twice differentiable function...
Text Solution
|
If f(x) satisfies the relation f((5x-3y)/2)=(5f(x)-3f(y))/2AAx ,y in ...
Text Solution
|
A function f: R rarr R" satisfies sin x cos y "(f(2x+2y)-f(2x-2y))= co...
Text Solution
|