Concept:For a function to have an extreme point at x=a, its first derivative must vanish at x=a.Explanation:Given f(x)=αlog∣x∣+βx2+x.Differentiating with respect to x, we get:f′(x)=xα+2βx+1.Since x=−1 and x=2 are extreme points, we have f′(−1)=0 and f′(2)=0.At x=−1:f′(−1)=−1α+2β(−1)+1=0⇒−α−2β+1=0⇒α+2β=1. ...(i)At x=2:f′(2)=2α+2β(2)+1=0⇒2α+4β+1=0⇒α+8β=−2. ...(ii)Subtracting equation (i) from equation (ii):(α+8β)−(α+2β)=−2−1⇒6β=−3⇒β=−21.Substitute β=−21 into equation (i):α+2(−21)=1⇒α−1=1⇒α=2.Answer:α=2, β=−21, which corresponds to option C.