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Question

Let f(x) and f(x) be differentiable for 0x1, such that f(0)=2,g(0)=0,f(1)=6. Let there exist a real number c in [0,1] such that f(c)=2g(c), then the value of g(1) must be

A
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2
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D
1
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Solution

The correct option is D 2
Given that f(0)=2,f(1)=6,g(0)=0

Given f(x),f(x) are differentiable on (0,1).

Therefore, Lagrange's mean value theorem can be applied.

Lagrange's mean value theorem states that if f(x) be continuous on [a,b] and differentiable on (a,b) then there exists some c between a and b such that f(c)=f(b)f(a)ba

Given that f(c)=2g(c)

f(1)f(0)10=2(g(1)g(0)10)

62=2(g(1)0)

2g(1)=4

g(1)=2

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