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Question

If p,q,r are real numbers, then f(x)=∣ ∣ ∣x+p2pqprpqx+q2qrprqrx+r2∣ ∣ ∣

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)x2
f(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)>0
x<23(p2+q2+r2) or x>0
For decreasing f'(x)<0
x<0 and 3x+2(p2+q2+r2)>0

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