To determine the IUPAC name:
1. Identify the longest continuous carbon chain, which has 8 carbons (octane).
2. Number the carbon chain to give the substituents the lowest possible locants: - Methyl groups are at positions 2, 5, and 6.
3. Arrange the substituents in alphabetical order and include their positions.
The compound is named : 2,5,6-Trimethyloctane
If \[ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6 \], then f(1) is equal to:
If the system of equations \[ (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \] \[ \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \] \[ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \] has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to: