Question

The set of all real values of \( a \) such that the function \( f(x) = x^3 + 2ax^2 + 3(a+1)x + 5 \) is strictly increasing in its entire domain is:

• A. \( (-\infty, -\frac{3}{4}) \cup (3, \infty) \)
• B. \( \left( -\frac{3}{4}, 3 \right) \)
• C. \( (1,3) \)
• D. \( (-\infty,1) \cup (3,\infty) \)

Hint:

For strictly increasing functions, check if the derivative is always positive by analyzing the discriminant of the quadratic inequality.

Answer 1

The Correct Option is B

Step 1: Compute the first derivative For \( f(x) \) to be strictly increasing, its first derivative must be positive for all \( x \): \[ f'(x) = 3x^2 + 4ax + 3(a+1). \] Step 2: Ensure positivity of \( f'(x) \) The quadratic expression \( 3x^2 + 4ax + 3(a+1)0 \) must be always positive, meaning its discriminant must be negative: \[ \Delta = (4a)^2 - 4(3)(3a+3)0. \] Solving for \( a \): \[ 16a^2 - 36a - 360. \] Factoring and solving the inequality, we find the valid range: \[ \left( -\frac{3}{4}, 3 \right). \]