Home
Class 12
MATHS
Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for a...

Let `(f(x+y)-f(x))/2=(f(y)-a)/2+x y` for all real `xa n dydot` If `f(x)` is differentiable and `f^(prime)(0)` exists for all real permissible value of `a` and is equal to `sqrt(5a-1-a^2)dot` Then `f(x)` is positive for all real `x` `f(x)` is negative for all real `x` `f(x)=0` has real roots Nothing can be said about the sign of `f(x)`

A

f(x) is positive for all real x

B

f(x) is negative for all real x

C

f(x)=0 has real roots

D

nothing can be said about the sign of f(x)

Text Solution

Verified by Experts

The correct Answer is:
A
We have `(f(x+y)-f(x))/(2)=(f(y)-a)/(2)+xy`
`therefore" "f(x+y)=f(x)+f(y)-a+2xy`
Put x = y = 0
`rArr" "f(0)=a`
`f'(x)=underset(hrarr0)(lim)(f(x+h)-f(x))/(h)`
`=underset(hrarr0)(lim)(f(x)+f(h)-a+2xh-f(x))/(h)`
`=underset(hrarr0)(lim)(f(h)-a)/(h)+2x`
`=underset(hrarr0)(lim)(f(h)-f(0))/(h)+2x`
`=f'(0)+2x`
`rArr" "f'(x)=2x+sqrt(5a-1-a^(2))`
`rArr" "f(x)=x^(2)+(sqrt(5a-1-a^(2)))x+c`
`rArr" "f(x)=x^(2)+(sqrt(5a-1-a^(2)))x+c`
Putting x = 0, we get c = a
`rArr" "f(x)=x^(2)+(sqrt(5a-1-a^(2)))x+a`
`=(x+(sqrt(5a-1-a^(2)))/(2))^(2)+a-(5a-1-a^(2))/(4)`
`=(x+(sqrt(5a-1-a^(2)))/(2))^(2)+(4a-5a+1+a^(2))/(4)`
`=(x+(sqrt(5a-1-a^(2)))/(2))^(2)+(a^(2)-a+1)/(4)gt0`
`" "(as a^(2)-a+1gt0)`
`therefore" "f(x)gt0 AA x in R`
Promotional Banner

Topper's Solved these Questions

  • METHODS OF DIFFERENTIATION

    CENGAGE|Exercise Multiple Correct Answer Type|7 Videos

  • MATRICES

    CENGAGE|Exercise Multiple Correct Answer|7 Videos

  • METHODS OF DIFFERETIATION

    CENGAGE|Exercise Question Bank|13 Videos

Similar Questions

Explore conceptually related problems

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real x and ydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then a) f(x) is positive for all real x b) f(x) is negative for all real x c) f(x)=0 has real roots d) Nothing can be said about the sign of f(x)

Let f(x+y)=f(x)+f(y)+2x y-1 for all real xa n dy and f(x) be a differentiable function. If f^(prime)(0)=cosalpha, then prove that f(x)>0AAx in Rdot

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot

Let f(x+y)=f(x)+f(y) for all xa n dydot If the function f(x) is continuous at x=0, show that f(x) is continuous for all xdot

If f(x)=(f(x))/y+(f(y))/x holds for all real x and y greater than 0a n df(x) is a differentiable function for all x >0 such that f(e)=1/e ,t h e nfin df(x)dot

H(x) increase as f(x) decreases for all real values of x if

If f((x+y)/3)=(2+f(x)+f(y))/3 for all x,y f'(2)=2 then find f(x)

Let f(x y)=f(x)f(y)AAx , y in Ra n df is differentiable at x=1 such that f^(prime)(1)=1. Also, f(1)!=0,f(2)=3. Then find f^(prime)(2)dot

If a function f satisfies f (f(x))=x+1 for all real values of x and if f(0) = 1/2 then f(1) is equal to

If f(x)= |x|^(3) , show that f''(x) exists for all real x and find it.