Explanation
To determine the value of q, we analyze the continuity of f(x) and the continuity of its derivative f′(x) at x=3.
Step 1: Continuity at x=3
For f(x) to be continuous at x=3, the limit as x approaches 3 from the left must equal the limit as x approaches 3 from the right:
x→3−lim(x−1)=x→3+lim(px2+qx+2)
3−1=p(3)2+q(3)+2
2=9p+3q+2
9p+3q=0⟹q=−3p…(Equation 1)
Step 2: Continuity of f′(x) at x=3
The derivative f′(x) is continuous at x=3 if the left-hand derivative equals the right-hand derivative:
x→3−limf′(x)=x→3+limf′(x)
For 1 < x < 3, f′(x)=dxd(x−1)=1.
For x > 3, f′(x)=dxd(px2+qx+2)=2px+q.
1=2p(3)+q
6p+q=1…(Equation 2)
Step 3: Solve for q
From Equation 1, we have p=−3q. Substitute this into Equation 2:
6(−3q)+q=1
−2q+q=1
−q=1
q=−1
Thus, the value of q is −1, and the correct option is (a).
