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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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CENGAGE PUBLICATION-CONTINUITY AND DIFFERENTIABILITY-All Questions
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  2. Find x where f(x) = max {sqrt(x(2-x)),2-x} is non-differentiable.

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  3. Discuss the differentiability of function f(x)=x-|x-x^2|

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  4. Discuss the differentiability of f(x)="sin"|x|

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  5. Discuss the continuity and differentiablity of f(x)=|x+1|+|x|+|x-1|...

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  7. find the values of a and b , such that f(x) ={ax^2+1,xle1 and x^2+ax...

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  8. Discuss the differentiability of f(x)=|x[x]| ,-1 < x <= 2

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  9. find the values of a and b , such that f(x) ={ax^2+1,xle1 and x^2+ax...

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  10. Discuss the differentiability of f(x)=cos^(-1)((1-x^2)/(1+x^2))

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  11. Which of the following function is non-differentiable in domain? (a)f(...

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  12. Discuss the continuity of f(x)={1/(e^(x-1)),x!=1; 0,x=1

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  13. Which of the following functions is not continuous AAx in R ?

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  14. Find the value of f(0) so that the function. f(x)=(sqrt(1+x)-1+x^3)/x ...

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  15. If the function f(x)={((x^2-(A+2)x+A)/(x-2), "for" x ne 2),(2, "for"...

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  16. If the function f: R- {0}-> R given by f(x)=1/x-2/(e^(2x)-1) is conti...

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  17. L e t f(x)=(1-tanx)/(4x-pi),x!=pi/4,x in [0,pi/2], Iff(x)i s continuou...

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  18. If f(x)=(tan(pi/4+(log)e x))^((log)x e) is to be made continuous at x=...

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  19. f(x)={(2xtanx-pi/(cosx) ,, x!=pi/2),(k ,, x=pi/2):} i s \ c o n t i n ...

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  20. Discuss the continuity of f(x) ={{:(((x^(2))/|x|,xne0),(0,x=0):}

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  21. Let f(x)={(1+3x)^(1/x), x != 0e^3,x=0 Discuss the continuity of f(x) a...

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