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Question

# If p,q,r are real numbers, then f(x)=âˆ£âˆ£ âˆ£ âˆ£âˆ£x+p2pqprpqx+q2qrprqrx+r2âˆ£âˆ£ âˆ£ âˆ£âˆ£

A
increases in x<(23(p2+q2+r2)) or x>0
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B
decreases in (23(p2+q2+r2),0)
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C
decreases in x<(23(p2+q2+r2)) or x>0
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D
increases in (23(p2+q2+r2),0)
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Solution

## The correct options are A increases in x<(−23(p2+q2+r2)) or x>0 B decreases in (−23(p2+q2+r2),0)f(x) =∣∣ ∣ ∣∣x+p2pqprpqx+q2qrprqrx+r2∣∣ ∣ ∣∣⇒f(x)=x3+(p2+r2+q2)x2f′(x)=3x2+2x(p2+q2+r2)=x(3x+2(p2+q2+r2))For increasing f'(x)>0⇒x>0 and 3x+2(p2+q2+r2)>0x<−23(p2+q2+r2) or x>0For decreasing f'(x)<0⇒x<0 and 3x+2(p2+q2+r2)>0

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