Question

If \[ f(x)=xe^{x^{2}-2x-3} \] is a real valued function, then \(f\) is

• A. a monotonically decreasing function
• B. an increasing function in the interval \((-1,3)\) and decreasing function in the interval \((-\infty,-1)\cup(3,\infty)\)
• C. a monotonically increasing function
• D. a decreasing function in the interval \((-1,3)\) and increasing function in the interval \((-\infty,-1)\cup(3,\infty)\)

Hint:

Whenever \[ f'(x) = (\text{positive function})\times(\text{quadratic}), \] first check whether the quadratic is always positive by completing the square. If \[ f'(x)>0 \] for all \(x\), then the function is monotonically increasing.

Answer 1

The Correct Option is C