Question

Find the local maximum point of the function \[ f(x)=-x^3+3x+1 \]

• A. \( x=1 \)
• B. \( x=-1 \)
• C. \( x=0 \)
• D. \( x=\sqrt3 \)

Hint:

A negative second derivative indicates downward concavity, which corresponds to a local maximum.

Answer 1

The Correct Option is A

Concept: Critical points occur where: \[ f'(x)=0 \] The second derivative test determines whether the point is a maximum or minimum: \[ f''(x)<0 \Rightarrow \text{Local Maximum} \] \[ f''(x)>0 \Rightarrow \text{Local Minimum} \]

Step 1: Differentiate the function:
\[ f(x)=-x^3+3x+1 \] \[ f'(x)=-3x^2+3 \]

Step 2: Find the critical points:
Set: \[ -3x^2+3=0 \] \[ x^2=1 \] \[ x=\pm1 \]

Step 3: Apply second derivative test:
\[ f''(x)=-6x \] At \(x=1\): \[ f''(1)=-6<0 \] Hence \(x=1\) is a local maximum point. At \(x=-1\): \[ f''(-1)=6>0 \] Hence \(x=-1\) is a local minimum point.