The value of the parameter 'a' such that the area bounded by $y = a[x^2] + ax + 1$, coordinate axes and the line $x = 1$ attains its least value, is equal to
Step-by-Step Solution
Key Concept: To find the area between two curves, integrate the difference of the upper and lower functions over the interval between their intersection points.
First, find the points of intersection by solving $\frac{1}{1+x^2} = x^2$, which gives $x^4 + x^2 - 1 = 0$, or $(x^2 - 1)(x^2 + 2) = 0$. This yields $x = \pm 1$ with $y = 1$. The area between curves is: $\text{Area} = 2\int_0^1 \left(\frac{1}{1+x^2} - x^2\right) dx = 2\left[\arctan(x) - \frac{x^3}{3}\right]_0^1 = 2\left(\frac{\pi}{4} - \frac{1}{3}\right) = \frac{\pi}{2} - \frac{2}{3}$ sq. units.
Correct Answer: 1