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Question

Given that the function f(x)=(x−p)2+(x−q)2+(x−r)2 has a minimum, find the corresponding values of x

A
13(p+q+r)
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B
23(p+q+r)
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C
12(p+q+r)
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D
None of these
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Solution

The correct option is B 13(p+q+r)
We have f(x)=(xp)2+(xq)2+(xr)2 ...(1)

It is differentiable function for x(,).

Hence by Fermat theorem, a minimum can be attained when f(x)=0

Differentiating (1) w.r.t x, we get

f(x)=2(xp)+2(xq)+2(xr) ..(2)

f(x)=0 when 2(xp)+2(xq)+2(xr)=0

3x(p+q+r)=0x=13(p+q+r)

Since a single value of x is obtained, without further investigation we can say the minimum is attained at x=13(p+q+r)

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