Let the function, f(x)={−3ax2−2,a2+bx,x<1x≥1 be differentiable for all x∈R, where a>1,b∈R. If the area of the region enclosed by y=f(x) and the line y=−20 is α+β3, where α,β∈Z, then the value of α+β is ____.
Answer: 34
Solution
For f to be differentiable at x=1, it must first be continuous there, and the one-sided derivatives must match.
Continuity: −3a(1)2−2=a2+b(1)⟹−3a−2=a2+b.
Differentiability: derivative of −3ax2−2 is −6ax, giving −6a at x=1; derivative of a2+bx is b. So b=−6a.
Substituting into the continuity condition:
−3a−2=a2−6a⟹a2−3a+2=0⟹(a−1)(a−2)=0.
Since a>1, a=2, and b=−12.
So
f(x)={−6x2−2,4−12x,x<1x≥1.
Find where each branch meets y=−20: Left branch: −6x2−2=−20⟹x2=3⟹x=−3 (the relevant root, since 3>1 lies outside x<1). Right branch: 4−12x=−20⟹x=2.
The enclosed region runs from x=−3 to x=2, bounded above by y=f(x) and below by y=−20:
