Question

The function $f(x) = x^3 + ax^2 + bx + c, a^2 \le 3b$ has

• A. one maximum value
• B. one minimum value
• C. no extreme value
• D. one maximum and one minimum value

Hint:

A cubic function is strictly monotonic if the discriminant of its derivative is $\le 0$.

Answer 1

The Correct Option is C

Step 1: Find Critical Points
$f'(x) = 3x^2 + 2ax + b$. Set $f'(x) = 0 \implies 3x^2 + 2ax + b = 0$.
Step 2: Analyze Roots of $f'(x)$
For extreme values to exist, $f'(x)=0$ must have real roots. Discriminant $D = (2a)^2 - 4(3)(b) = 4a^2 - 12b = 4(a^2 - 3b)$.
Step 3: Check Given Condition
Given $a^2 \le 3b \implies a^2 - 3b \le 0$. This means $D \le 0$, so $f'(x)$ has no distinct real roots. If $f'(x)$ never changes sign, the function is monotonic and has no maximum or minimum values.
Final Answer: (c)