Question

The function $f(x) = \begin{vmatrix} x^{2} & x \\ 3 & 1 \end{vmatrix}, x \in \mathbb{R}$ has:

• A. local minimum at \(x=\frac{3}{2}\)
• B. local maximum at \(x=\frac{3}{2}\)
• C. local minimum at \(x=0\)
• D. local minimum at \(x=0\)

Hint:

Use determinant expansion first, then use calculus to find maxima or minima.

Answer 1

The Correct Option is B

Step 1: Evaluate the determinant function: \[ f(x) = \begin{vmatrix} x^{2} & x \\ 3 & 1 \end{vmatrix} = x^{2} \cdot 1 - x \cdot 3 = x^{2} - 3x \] Step 2: Find critical points by setting derivative zero: \[ f'(x) = 2x - 3 = 0 \implies x = \frac{3}{2} \] Step 3: Find second derivative to classify the critical point: \[ f''(x) = 2 \] Since \(f''(x) = 2 > 0\), the function has a local minimum at \(x = \frac{3}{2}\).