At extreme points, the derivative equals zero.
Given: f(x)=αlog∣x∣+βx2+x
The derivative of log∣x∣ is x1
f′(x)=xα+2βx+1
Since x=−1 is an extreme point, f′(−1)=0:
−1α+2β(−1)+1=0
−α−2β+1=0
α+2β=1 ...(Equation 1)
Since x=−2 is an extreme point, f′(−2)=0:
−2α+2β(−2)+1=0
−2α−4β+1=0
Multiplying by 2:
−α−8β+2=0
α+8β=2 ...(Equation 2)
From Equation 1: α+2β=1
From Equation 2: α+8β=2
Subtracting Equation 1 from Equation 2:
6β=1
β=61
Substituting into Equation 1:
α+2(61)=1
α+31=1
α=32
Therefore, α=32 and β=61