Practicing Success
Let f be a positive function, and let $I_1 = \int\limits_{1-k}^{k}xf[x(1-x)]dx$, $I_2 = \int\limits_{1-k}^{k}f[x(1-x)dx]$, $k >\frac{1}{2}$. If $I_1 = t\, I_2$, then t is equal to |
2 k 1/2 1 |
1/2 |
We have $k >\frac{1}{2}$ ⇒ 2k - 1 > 0 ⇒ k > 1 - k. Hence $I_1 = \int\limits_{1-k}^{k}xf(x(1-x))dx=\int\limits_{1-k}^{k}(k+1-k-x)f(k+1-k-x)(1-k-1+k+x)dx$. $\int\limits_{1-k}^{k}(1-x)f(x(1-x))dx$ ⇒ I1 = I2 - I1 or, 2I1 = I2 ⇒ $t=\frac{1}{2}$ Hence (C) is the correct answer. |