Question

If \( x = -0.5 \), then which of the following has the smallest value?

• A. \( 2^x \)
• B. \( 1/x \)
• C. \( 1/x^2 \)
• D. \( 2x^x \)
• E. \( 1/\sqrt{-x} \)

Hint:

For questions asking for the "smallest" value with negative inputs, look for expressions that result in negative numbers first, as they will always be smaller than positive results.

Answer 1

The Correct Option is B

Step 1: Understanding the value of \( x \): 
We are given that \( x = -0.5 \). Let’s evaluate the value of each expression. 
Step 2: Evaluating each option: 
(a) \( 2^x = 2^{-0.5} = \frac{1}{\sqrt{2}} \approx 0.707 \) 
(b) \( \frac{1}{x} = \frac{1}{-0.5} = -2 \) 
(c) \( \frac{1}{x^2} = \frac{1}{(-0.5)^2} = \frac{1}{0.25} = 4 \) 
(d) \( 2x^x = 2(-0.5)^{-0.5} = 2 \times \frac{1}{\sqrt{-0.5}} \) 
(This is undefined because we cannot take the square root of a negative number. Hence, this option is not valid.) 
(e) \( \frac{1}{\sqrt{-x}} = \frac{1}{\sqrt{0.5}} \approx 1.414 \) 
Step 3: Comparing the results: 
Now, let's compare the results of the valid options: 
- (a) \( 2^x \approx 0.707 \) 
- (b) \( \frac{1}{x} = -2 \) 
- (c) \( \frac{1}{x^2} = 4 \) 
- (e) \( \frac{1}{\sqrt{-x}} \approx 1.414 \)
Step 4: Conclusion: 
The smallest value among the valid options is \( \frac{1}{x} = -2 \). Thus, the correct answer is \( \boxed{(b)} \).