Question

Let \( f(x) = x^3 - \frac{9}{2}x^2 + 6x - 2 \) be a function defined on the closed interval [0, 3]. Then, the global maximum value of \( f(x) \) is _______}

• A. 4.5
• B. 0.5
• C. 2.5
• D. 3.0

Hint:

For finding global maxima or minima, evaluate the function at the critical points and endpoints of the interval, and compare the results.

Answer 1

The Correct Option is C

To find the global maximum value of the function on the closed interval [0, 3], we first compute the derivative of \( f(x) \):
\[ f'(x) = 3x^2 - 9x + 6 \] We then solve for critical points by setting \( f'(x) = 0 \) and solving for \( x \):
\[ 3x^2 - 9x + 6 = 0 \] Solving the quadratic equation, we get the critical points.
Next, we evaluate \( f(x) \) at the critical points and at the endpoints of the interval \( x = 0 \) and \( x = 3 \).
After evaluating, we find that the maximum value of \( f(x) \) is \( 2.5 \). Hence, the correct answer is \( 2.5 \).