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Let f(x)=2x+3" " for xle1 =ax...

Let `f(x)=2x+3" "` for `xle1`
`=ax^(2)+bx" "` for `xgt1`
If f(x) is everywhere differentiable, then prove that `f'(2)=-4`.

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CHHAYA PUBLICATION-MISCELLANEOUS EXAMPLES-CALCULUS
  1. Examine the differentiability of f(x)=|sinx-cosx|" at "x=(pi)/(4)

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  2. Prove that the function f(x)=|x| is not differentiable there at x=0.

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  3. Let f(x)=2x+3" " for xle1 =ax^(2)+bx" " for xgt1 If...

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  4. If f(2)=4,g(2)=9andf'(2)=2g'(2), then evaluate lim(xto2)(sqrt(f(x))-2)...

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  5. If f(9)=9,f'(9)=4, then evaluate lim(xto9)(sqrtf(x)-3)/(sqrtx-3).

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  6. Let f((x+y)/(2))=(1)/(2)[f(x)+f(y)] for all real x and y. If f'(0) exi...

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  7. If f(x) is differentiable at x=a, find the value of lim(xtoa)((x+a)f(x...

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  8. The value of lim(xto0)f(x) where f(x)=(x^(3)+x^(2))/(2x^(3)-27x^(2))

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  9. If f(x+y)=f(x)f(y) for all x, y and f(x)=1+xg(x), where lim(xto0)g(x)=...

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  10. If f(x+y+z)=f(x)f(y)f(z)ne0, for all x, y, z and f(2)=4,f'(0)=3, find ...

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  11. If f(3)=6andf'(3)=2, find the value of lim(xto3)(xf(3)-3f(x))/(x-3).

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  12. If y=f{f(x)},f(0)=0andf'(0)=5, find [(dy)/(dx)](x=0)

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  13. Find the derivative of |f(x)| with respect to x, hence, write down the...

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  14. Prove that, lim(ntooo)e^((2)/(n)+1)=e

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  15. Prove that, lim(ntooo)[(n)/(n^(2))+(n+1)/(n^(2))+(n+2)/(n^(2))+........

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  16. Prove that, lim(ntooo)(3n^(3)-5n^(2)+4)/(5+3n^(2)-4n^(3))=-(3)/(4)

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  17. Prove that, lim(xtooo)(4x^(3)-5x^(2)+6x+9)/(3x^(4)+4x^(2)-11)=0

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  18. Prove that, lim(ntooo)(1^(2)+2^(2)+3^(2)+.......+(n+2)^(2))/(n^(3))=...

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  19. Prove that, lim(xtooo)((1)/(4)+(1)/(4^(2))+(1)/(4^(3))+.......+(1)/(...

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  20. Prove that, lim(xtooo)(1+2+3+......"to" (4x+1)"terms")/((x+1)^(2))=8

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