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. \(b^2\lt 3ac\)
• D. \(b^2\gt 3ac\)

Hint:

A cubic polynomial has extreme values only when its derivative has real roots.

Answer 1

The Correct Option is C

Step 1: Differentiate the function.
The given function is \[ f(x)=ax^3+bx^2+cx+d \] Differentiate with respect to \(x\): \[ f'(x)=3ax^2+2bx+c \] Extreme values occur when \[ f'(x)=0 \] Thus, \[ 3ax^2+2bx+c=0 \]

Step 2: Condition for no extreme value.
For the cubic polynomial to have no maximum or minimum value, the quadratic equation \[ 3ax^2+2bx+c=0 \] must have no real roots.
A quadratic equation has no real roots when its discriminant is negative.

Step 3: Find the discriminant.
The discriminant is \[ D=(2b)^2-4(3a)(c) \] \[ =4b^2-12ac \] \[ =4(b^2-3ac) \] For no real roots, \[ D\lt 0 \] Thus, \[ 4(b^2-3ac)\lt 0 \] \[ b^2-3ac\lt 0 \] \[ b^2\lt 3ac \]

Step 4: Final Answer.
Therefore, the required condition is \[ \boxed{b^2\lt 3ac} \]