Question

The condition that \( f(x) = ax^3 + bx^2 + cx + d \) has no extreme value is

• A. \( b^2 - 4ac \)
• B. \( b^2 = 3ac \)
• C. \( b^2<3ac \)
• D. \( b^2>3ac \)

Hint:

For a cubic to have no extreme values, its first derivative must have no real roots — check the discriminant!

Answer 1

The Correct Option is C

To find extreme values of a cubic function \( f(x) = ax^3 + bx^2 + cx + d \), we first find \( f'(x) = 3ax^2 + 2bx + c \).
For extreme values to exist, this quadratic must have real roots.
So, check discriminant: \( D = (2b)^2 - 4 \cdot 3a \cdot c = 4b^2 - 12ac \).
If there are no real roots, the derivative never becomes zero \( \Rightarrow \) no extreme value.
So, the condition for no extreme value is \( 4b^2 - 12ac<0 \Rightarrow b^2<3ac \).