Question

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

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

Hint:

For cubic \(f(x)\), extremes occur only when \(f'(x)\) has two real distinct roots i.e. discriminant \(>0\).

Answer 1

The Correct Option is C

Step 1: Find first derivative.
\[ f'(x)=3x^2+2ax+b \] 
Step 2: Condition for extreme values. 
Extreme values exist if \(f'(x)=0\) has two distinct real roots. 
That depends on discriminant of quadratic \(3x^2+2ax+b\). 
Step 3: Compute discriminant. 
\[ \Delta=(2a)^2-4(3)(b)=4a^2-12b=4(a^2-3b) \] 
Step 4: Use given condition. 
Given: \(a^2\leq 3b\). 
So: 
\[ a^2-3b\leq 0 \Rightarrow \Delta\leq 0 \] 
Step 5: Conclusion. 
If \(\Delta<0\), no real critical points \(\Rightarrow\) no maxima/minima. 
If \(\Delta=0\), only one stationary point (point of inflection), still no max/min. 
Thus function has no extreme value. 
Final Answer: 
\[ \boxed{\text{no extreme value}} \]