Question

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

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

Hint:

For a continuous strictly increasing function, the range is $[f(min), f(max)]$.

Answer 1

The Correct Option is B

Step 1: Differentiation
$f'(x) = 3x^{2} + 3$.
Step 2: Analysis of Monotonicity
Setting $f'(x) = 0$ gives $x^{2} = -1$, which has no real roots. This means $f(x)$ is strictly increasing for all real $x$.
Step 3: Calculating Endpoints
At $x = 2$, $f(2) = 2^3 + 3(2) - 2 = 8 + 6 - 2 = 12$.
At $x = 3$, $f(3) = 3^3 + 3(3) - 2 = 27 + 9 - 2 = 34$.
Step 4: Range
Since the function is strictly increasing, the range is $[12, 34]$.
Final Answer: (b)