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A function f:R rarr R satisfies the equa...

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. Prove that f'(x)=2f(x).

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PATHFINDER-LIMIT, CONTINUITY AND DIFFERENTIABILITY-QUESTION BANK
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  2. The left hand derivative of f(x)=[x]sin (pix) at x=k, k is an integer,...

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  3. A function f:R rarr R satisfies the equation f(x+y)=f(x)f(y) for all x...

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  4. Let f(x)={(-4, -4 lexlt0),(x^2-4, 0lexle4):} Discuss the continuity a...

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  5. Let f(x)=[n+p sin x], x in (0, pi) n in Z and p is a prime number, whe...

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  6. If f(x)={|x|-|x-1|}^2, draw the graph of f(x) and discuss its continui...

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  7. If f(x)=|x+1|{|x|+|x-1|}, then draw the graph of f(x) in the interval ...

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  8. The value of lim(xrarr0)(int0^(x^2) cos(t^2)dt)/(xsinx) is

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  9. The value of lim(xrarr0)(1^x+2^x+3^x+...+n^x)^(a//x)/n, is:

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  10. If lim(xrarr0)(1+ax+bx^2)^(2//x)=e^3, then the value of a and b, is :

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  11. A function is defined as, f(x)={(0, where x is rational),(1, where x ...

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  12. If f(x)={(-1, xlt0 ),(0, x =0),(1, x gt 0):} and g(x)=x(1-x^2), then...

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  13. If f(x)=-1+|x-2|, 0lexle4 g(x)=2-|x|, -1lexle3 Then, fog(x) is contin...

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  14. Let f(x)=[sinx + cosx], 0ltxlt2pi, (where [.] denotes the greatest int...

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  15. If f(x)={(sin{cosx}/(x-pi/2), x ne pi/2),(1, x=pi/2):} , where {.} den...

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  16. If f(x)=(x^3+x^2-16x+20)/(x-2)^2, x ne 2 =k, x=2 and if f(x) is con...

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  17. In the function f(x)=[(x-2)^3/a]sin(x-2)+acos(x-2), (where [.] denote...

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  18. Let f(x) = |x|+|sin x|, x in (-pi/2, (3pi)/2). Then, f is :

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  19. If f is a periodic function, then

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  20. If f(x)=[sin^2x] (where [.] denotes the greatest integer function ) th...

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