Question

Find the maximum value of f(x) = (1 / (4x² + 2x + 1)).

• A. \( \frac{3}{4} \)
• B. \( \frac{4}{3} \)
• C. \( \frac{1}{3} \)
• D. None of these

Hint:

To find the maximum or minimum value of a rational function, differentiate the function, set the first derivative equal to zero, and check the behavior of the function at critical points.

Answer 1

The Correct Option is B

To find the maximum value of \( f(x) = \frac{1}{4x^2 + 2x + 1} \), we first differentiate \( f(x) \). 1. The function is \( f(x) = \frac{1}{4x^2 + 2x + 1} \). 2. Differentiate \( f(x) \) with respect to \( x \) using the chain rule: \[ f'(x) = \frac{- (8x + 2)}{(4x^2 + 2x + 1)^2} \] 3. Set \( f'(x) = 0 \) to find critical points. This occurs when the numerator is zero, i.e., \( 8x + 2 = 0 \), solving for \( x \), we get \( x = -\frac{1}{4} \). 4. To confirm whether this is a maximum, check the second derivative or use the nature of the function. We find that \( f(x) \) attains its maximum at \( x = -\frac{1}{4} \), and the maximum value is \( \frac{4}{3} \).