The correct option is(C): 3-Hydroxy-4 -methylpentanoic acid
The given compound is a carboxylic acid with a hydroxy group and a methyl group attached to its carbon chain. To determine its IUPAC name, let's break down the structure and identify the substituents:
Putting it all together, the IUPAC name of the compound is "3-Hydroxy-4-methylpentanoic acid."
Justification:
Let $ f(x) = \begin{cases} (1+ax)^{1/x} & , x<0 \\1+b & , x = 0 \\\frac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2} & , x>0 \end{cases} $ be continuous at x = 0. Then $ e^a bc $ is equal to