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Question

Let f(x)=xp(sinx)q,if 0<xπ20,if x=0,(p,q,ϵR). Then Lagrange's mean value theorem is applicable to f(x) in closed interval [0,x].

A
For all p,q
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B
Only when p>q
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C
Only when p<q
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D
For no value of p,q
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Solution

The correct option is B Only when p>q
f(x)=xp(sinx)q,if 0<xπ20,if x=0,(p,q,ϵR)
Lagrange's Mean value theorem is applicable for any f, if it is continuous at (a,b) and differential with in the closed interval [a,b].
i.e. if limxaf(x)=f(a)
Here, f(a)=f(0)=0
f(x) is differentiable at (0,x)
Consider, limx0f(x)
=limx0xp(sinx)q

=limx0xpxqxq(sinx)q

=limx0xqxpq(sinx)q

=limx0xq(sinx)q×limx0xpq

=limx0(xsinx)q×limx0xpq

=limx0xpq ....... [limx0xsinx=1]
Now, limx0xpq=0,if p>q,if p<q1,if p=q
limx0f(x)=0 only when p>q
Hence, Lagrange's Mean value theorem is applicable to f(x) in [0,x] only when p>q.

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