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: Calculate the determinant of the function: \[ f(x) = \begin{vmatrix} x^{2} & x \\ 3 & 1 \end{vmatrix} = x^{2} \cdot 1 - x \cdot 3 = x^{2} - 3x \] Step 2: Identify critical points by equating the derivative to zero: \[ f'(x) = 2x - 3 = 0 \implies x = \frac{3}{2} \] Step 3: Determine the nature of the critical point using the second derivative: \[ f''(x) = 2 \] As \(f''(x) = 2>0\), the function exhibits a local minimum at \(x = \frac{3}{2}\).