Question

What is the least value of \( k \) such that the function \( x^2 + kx + 1 \) is strictly increasing on \( (1, 2) \)?

• A. 1
• B. -1
• C. 2
• D. -2

Hint:

To determine if a function is increasing, find its derivative and ensure that it is positive on the desired interval.

Answer 1

The Correct Option is D

Step 1: Use the derivative. The function is strictly increasing if its derivative is positive. The derivative of \( f(x) = x^2 + kx + 1 \) is \( f'(x) = 2x + k \).
Step 2: Solve for \( k \). For the function to be increasing on \( (1, 2) \), \( f'(x) \) must be positive on this interval. Solving \( 2x + k>0 \) for \( k \), we get the least value \( k = -2 \).
Step 3: Conclusion. Thus, the least value of \( k \) is -2.