IMAT 2011 Q77 [Largest Value]

Which of the expressions below has the largest value for 0 < x < 1?

A. {\frac{1}{x}}

B. {x^2}

C. {\frac{1}{(1+x)}}

D. {\frac{1}{\sqrt x}}

E. {\sqrt x}

For solving this questions it is necessary to know the graphs of the functions (or draw them approximately).

Looking at the graphs for 0<x<1, we can see that the y=\frac{1}{x} and y=\frac{1}{\sqrt x} look similar.

The only valid explanation is that the y=\frac{1}{x} has bigger values for 0<x<1 than y=\frac{1}{\sqrt x}.
Let’s take x_1=\frac{1}{4} as an example in order to explain:

For the first one: y=\frac{1}{x_1} = \frac{1}{\frac{1}{4}} = 4

For the second one: y=\frac{1}{\sqrt{ x_1}} =\frac{1}{\sqrt{ \frac{1}{4}}} = \frac{1}{\frac{1}{2}} = 2.

Which proves that the values for (A) \frac{1}{x} are the largest amongst the group.