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Let alpha in R. Prove that a function...

Let `alpha in R`. Prove that a function `f: RtoR` is differentiable at `alpha` if and only if there is a function `g:RtoR` which is continuous at `alpha` and satisfies `f (x) -f(alpha) = g (x) (x-alpha), AA x in R.`

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The correct Answer is:
`x = alpha`
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ARIHANT MATHS-CONTINUITY AND DIFFERENTIABILITY-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Let alpha in R. Prove that a function f: RtoR is differentiable a...

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  2. For every pair of continuous function f, g : [0, 1] rarr R such that m...

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  3. Let f: R to R and g:R to R be respectively given by f(x) =|x|+1 and g...

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  4. Let f(x)={x^2|(cos)pi/x|, x!=0 and 0,x=0,x in RR, then f is a. differ...

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  5. Q. For every integer n, leta(n) and b(n) be real numbers. Let functio...

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  6. Let f:R->R be a function such that f(x+y)=f(x)+f(y),AA x, y in R. a. ...

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  7. If f(x) ={{:(-x=(pi)/(2),xle -(pi)/(2)), (- cos x, -(pi)/(2)lt x ,le ...

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  8. For the function f(x)=x cos ""1/x, x ge 1 which one of the following i...

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  9. If lim(xtooo)((x^(2)+x+1)/(x+1)-ax-b)=4, then

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  10. Let fandg be real valued functions defined on interval (-1,1) such tha...

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  11. If f(x)=x-[x], where [x] is the greatest integer less than or equal to...

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  12. Check the differentiability if f(x) = min. {1, x^(2), x^(3)}.

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  13. Fill in the blank, in the statement given below. Let f(x) = x|x The ...

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  14. lf is a differentiable function satisfying f(1/n)=0,AA n>=1,n in I, th...

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  15. The domain of the derivative of the function f(x)={{:(tan^(-1)x ,if|x|...

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  16. The left hand derivative of f(x) = [x] sin (pi x)"at x" = k is an inte...

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  17. Which of the following functions is differentiable at x = 0 ?

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  18. For x in R, f(x) =|log(e) 2-sinx| and g(x) = f(f(x)) , then

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  19. If the function g(X) ={{:( ksqrt ( x+1), 0 le x le 3),( mx+2, 3 lt x...

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  20. If f and g are differentiable functions in [0, 1] satisfying f(0)""=""...

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  21. The function f(x) = [x] cos((2x-1)/2) pi where [ ] denotes the greate...

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