Question

The function \(x^5-5 x^4+5 x^3-10\) has a maximum, when \(x\) is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

Find where the first derivative is zero and check the sign change of the derivative.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
A maximum occurs where \(f'(x)=0\) and \(f'\) changes from positive to negative.

Step 2: Differentiate
\(f'(x)=5x^{4}-20x^{3}+15x^{2}=5x^{2}(x-1)(x-3)\).
The critical points are \(x=0,1,3\).

Step 3: Check the Sign
Near \(x=1\): for \(x\) slightly less than 1, \(f'(x) \gt 0\); slightly more than 1, \(f'(x) \lt 0\). So \(x=1\) is a maximum.
At \(x=0\), \(f'\) does not change sign; at \(x=3\) it is a minimum.

Step 4: Result
The maximum is at \(x=1\), option (B).

Final Answer:
The function has a maximum at \(x=1\). \[ \boxed{\text{(B) }1} \]