Let the function,
f(x)= -3 ax²-2, x<1 ; a²+b x, x ≥ 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+β √3, α, β ∈ Z, then the value of α+β is ___________ .

Solution

Maths sir · 5 steps

Join the pieces smoothly at x = 1 to find a and b, then integrate the gap down to the line y = −20.

1. Match at the joint. Continuity at x = 1 gives -3a - 2 = a² + b. Equal slopes there give -6a = b, because the left piece has derivative -6ax and the right piece has derivative b.
2. Find a and b. Put b = -6a into the first equation to get a² - 3a + 2 = 0, so a = 1 or a = 2. Since a > 1, a = 2 and b = -12. Now f(x) = -6x² - 2 for x < 1 and f(x) = 4 - 12x for x ≥ 1.
3. Meet the line. On the left, -6x² - 2 = -20 gives x² = 3, so x = -√3. On the right, 4 - 12x = -20 gives x = 2. Between these two points the curve stays above y = -20.

   A = ∫_(-√3)¹ (18 - 6x²) dx + ∫₁² (24 - 12x) dx

4. Integrate. The first integral is [18x - 2x³]_(-√3)¹ = 16 - (-12√3) = 16 + 12√3. The second is [24x - 6x²]₁² = 24 - 18 = 6. This second piece is a triangle with base 1 and height 12.
5. So. A = 22 + 12√3, so α = 22 and β = 12, and α + β = 34.

The common slip: 28. Integrating only the parabola piece and leaving out the triangle from x = 1 to x = 2 gives 16 + 12√3, so α + β = 28, not 34.

The idea to keep. For a differentiable piecewise function, write two equations at the joint: one for equal values and one for equal slopes. Then use the given condition, like a > 1, to choose the right root.

Answer from the NTA final answer key. Solution by PrepWiser.