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If f(x) is continuous on [0,2] , differe...

If f(x) is continuous on [0,2] , differentiable in (0,2) ,f(0)=2, f(2)=8 and f.(x) `le` 3 for all x in (0,2), then find the value of f(1).

A

4

B

5

C

6

D

7

Text Solution

Verified by Experts

The correct Answer is:
B
Applying LMVT to f in [0,1] and again in [1,2] there exists `C_1 in (0,1)` such that
`(f(1)-f(0))/(1-0) = f.C_1 1 le 3 rArr f(1) le 5` …(i)
There exists `C_2 in (1,2)` , such that
`(f(2)-f(1))/(2-1) = f.(C_2) le 3 rArr f(1) ge 5`
Hence, (1) and (2) imply that f(1)=5
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Knowledge Check

  • If f (x) is continous on [0,2], differentiable in (0,2) f (0) =2, f(2)=8 and f '(x) le 3 for all x in (0,2), then find the value of f (1).

    A
    4
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    D
    7

  • Let the function, f:[-7,0]toR be continuous on [-7,0] and differentiable on (-7,0) . If f(-7)=-3 and f'(x)le2, for all x in(-7,0) , then for all such functions f,f(-1)+f(0) lies in the interval :

    A
    `[-6,20]`
    B
    `(-oo,20]`
    C
    `(-oo,11]`
    D
    `[-3,11]`

  • If f(x) is continuous in [0,2] and f(0)=f(2). Then the equation f(x)=f(x+1) has

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