Question

If \(f:[2,3]\rightarrow \mathbb{R}\) is defined by \(f(x)=x^3+3x-2\), then the range \(f(x)\) is contained in the interval

• A. \([1,12]\)
• B. \([12,34]\)
• C. \([35,50]\)
• D. \([-12,12]\)

Hint:

If \(f'(x)>0\) in an interval, \(f(x)\) is increasing, so range is \([f(a),f(b)]\).

Answer 1

The Correct Option is B

Step 1: Check monotonicity on \([2,3]\).
\[ f(x)=x^3+3x-2 \] 
Differentiate: 
\[ f'(x)=3x^2+3=3(x^2+1) \] 
Since \(x^2+1>0\) for all real \(x\), 
\[ f'(x)>0 \;\Rightarrow\; f(x)\text{ is strictly increasing on }[2,3] \] 
Step 2: Find minimum and maximum values. 
Minimum at \(x=2\): 
\[ f(2)=2^3+3(2)-2=8+6-2=12 \] 
Maximum at \(x=3\): 
\[ f(3)=3^3+3(3)-2=27+9-2=34 \] 
Step 3: Write the range. 
\[ \text{Range}=[12,34] \] 
Final Answer: 
\[ \boxed{[12,34]} \]