Questions: Identify all absolute extreme values of f(x)=x^(3/2) on the interval [4,5].

Solution Steps

To find the absolute extreme values of the function \( f(x) = x^{\frac{3}{2}} \) on the interval \([4, 5]\), we need to evaluate the function at the endpoints of the interval and at any critical points within the interval. Critical points occur where the derivative is zero or undefined. We will calculate the derivative, find any critical points, and then evaluate the function at these points and the endpoints to determine the absolute extrema.

We are given the function \( f(x) = x^{\frac{3}{2}} \) and the interval \([4, 5]\). Our goal is to find the absolute extreme values of this function on the specified interval.

The derivative of the function is calculated as follows: \[ f'(x) = \frac{3}{2} x^{\frac{1}{2}} = 1.5 x^{0.5} \] We set the derivative equal to zero to find critical points: \[ 1.5 x^{0.5} = 0 \implies x = 0 \] However, \( x = 0 \) is not within the interval \([4, 5]\).

Next, we evaluate the function at the endpoints of the interval: \[ f(4) = 4^{\frac{3}{2}} = 8 \] \[ f(5) = 5^{\frac{3}{2}} \approx 11.1803 \]

Since there are no critical points within the interval, we compare the function values at the endpoints:

• \( f(4) = 8 \)
• \( f(5) \approx 11.1803 \)

Thus, the absolute minimum value is \( 8 \) at \( x = 4 \), and the absolute maximum value is approximately \( 11.1803 \) at \( x = 5 \).

Final Answer