Question

Let f: ℝ → ℝ be defined by
\(f(x)=\frac{(x^2+1)^2}{x^4+x^2+1}\) for x ∈ \(\R\).
Then, which one of the following is TRUE ?

• A. f has exactly two points of local maxima and exactly three points of local minima
• B. f has exactly three points of local maxima and exactly two points of local minima
• C. f has exactly one point of local maximum and exactly two points of local minima
• D. f has exactly two points of local maxima and exactly one point of local minimum

Answer 1

The Correct Option is B

To find the local maxima and minima, we first need to compute the first derivative \( f'(x) \) and find the critical points where \( f'(x) = 0 \). The function is: \[ f(x) = \frac{(x^2 + 1)^2}{x^4 + x^2 + 1}. \] To simplify finding the critical points, we use the quotient rule: \[ f'(x) = \frac{(2x(x^2 + 1)(x^4 + x^2 + 1)) - (4x^3(x^2 + 1)^2)}{(x^4 + x^2 + 1)^2}. \] By solving \( f'(x) = 0 \), we find that the function has critical points. Analyzing the second derivative or examining the behavior of \( f(x) \) at those points reveals that there are two local maxima and one local minimum. Therefore, the correct answer is (D): \( f \) has exactly two points of local maxima and exactly one point of local minimum.