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 cubic.
For $f(x)=ax^3+bx^2+cx+d$, the derivative is $f'(x)=3ax^2+2bx+c$.
Step 2: Recall what an extreme value needs.
Extreme values occur where $f'(x)=0$ and the sign of $f'$ changes. So $f$ has extreme values only if $3ax^2+2bx+c=0$ has real roots.
Step 3: State the no extremum condition.
For no extreme value, this quadratic must have no real roots, meaning its discriminant is negative.
Step 4: Compute the discriminant.
$D=(2b)^2-4(3a)(c)=4b^2-12ac=4(b^2-3ac)$.
Step 5: Impose $D<0$.
$4(b^2-3ac)<0$ gives $b^2-3ac<0$.
Step 6: Simplify.
Therefore the condition is $b^2<3ac$.
\[ \boxed{b^2<3ac} \]