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Let f(x) =e^(x), x in [0,1] , then a num...

Let `f(x) =e^(x), x in [0,1]` , then a number c of the Largrange's mean value theorem is

A

`log_(e)(e-1)`

B

`log_(e)(e+1)`

C

`log_(e)e`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
Clearly, f(x) is continuous on [0,1] and differentiable on (0,1). Therefore, therer exists `c in (0,1)` such that
`f'(c)=(f(1)-f(0))/(1-0)rArr e^(c)=e-1rArr c=log_(e) (e-1)`
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OBJECTIVE RD SHARMA ENGLISH-MEAN VALUE THEOREMS-Exercise
  1. Let f(x) =e^(x), x in [0,1] , then a number c of the Largrange's mean ...

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  2. Let a and b be two distinct roots of a polynomial equation f(x) =0 The...

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  3. If 2a+3b+6c=0, then prove that at least one root of the equation a x^2...

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  4. Let f(x)a n dg(x) be two functions which are defined and differentiabl...

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  5. Let f be differentiable for all x , If f(1)=-2a n df^(prime)(x)geq2 fo...

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  6. If the function f(x)=x^3-6x^2+a x+b defined on [1,3] satisfies Rolles ...

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  7. Let (a0)/(n+1)+(a1)/n+(a2)/(n-1)++(a(n-1))/2+an=0. Show that there e...

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  8. The number of values of k for which the equation x^3-3x+k=0 has two di...

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  9. if f(x)=(x -4) (x-5) (x-6) (x-7) then, (A) f'(x) =0 has four roots (...

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  10. Let fa n dg be differentiable on [0,1] such that f(0)=2,g(0),f(1)=6a n...

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  11. If the equation a(n)x^(n)+a(n-1)x^(n-1)+..+a(1)x=0, a(1)!=0, n ge2, ha...

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  12. The equation x log x = 3-x has, in the interval (1,3) :

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  13. If f(x) and g(x) ar edifferentiable function for 0lex le1 such that f(...

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  14. If alpha beta( alpha lt beta) are two distinct roots of the equation. ...

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  15. If (x) is a function given by f(x) = |{:(sinx , sin a, sin b),(cosx,c...

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  16. The value of c in Lagrange's theorem for the functin f(x)=log sin x in...

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  17. n is a positive integer. If the value of c presecribed in Rolle's th...

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  18. The distance travelled by a particle upto tiem x is given by f(x)=x^(...

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  19. The number of real roots of the equation e^(x-1)+x-2=0 is 1 (b) 2 (...

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  20. If the polynomial equation an x^n + a(n-1) x^(n-1) + a(n-2) x^(n-2) + ...

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  21. If 4a+2b+c=0 , then the equation 3ax^(2)+2bx+c=0 has at least one rea...

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