Find the absolute extreme values of the function on the interval.
-$$F ( x ) = - \frac { 2 } { x ^ { 2 } } , 0.5 \leq x \leq 5$$
A) absolute maximum is $- \frac { 2 } { 25 }$ at $x = 5$; absolute minimum is $- 8$ at $x = \frac { 1 } { 2 }$
B) absolute maximum is $- \frac { 2 } { 25 }$ at $x = 5$; absolute minimum is $- 8$ at $x = - \frac { 1 } { 2 }$
C) absolute maximum is $\frac { 2 } { 25 }$ at $x = \frac { 1 } { 2 }$; absolute minimum is $- 8$ at $x = 5$
D) absolute maximum is $- \frac { 2 } { 25 }$ at $x = \frac { 1 } { 2 }$; absolute minimum is $- 8$ at $x = - 5$

Question

Quizplus · Accepted Answer

To find the absolute extreme values, we need to look for critical points in the interval, which are the points where the derivative is equal to zero or undefined, and the endpoints of the interval.

The derivative of the function is:

$$F'(x) = \frac{4}{x^3}$$

This is defined for all values of $x$ in the interval except $x=0$. But $0$ is outside of the interval, so we don't need to worry about it.

The derivative is equal to zero when $x=0$, but again, this is outside of the interval, so we don't need to worry about it.

Therefore, the only critical point in the interval is when $F'(x)$ is undefined, which occurs when $x=0.5$.

Now we need to compare the function value at this critical point and the endpoints of the interval to find the absolute extreme values:

$$F(0.5) = -8$$
$$F(5) = -\frac{2}{25}$$
$$F(0) \text{ and } F(5) \text{ are not defined}$$

Therefore, the absolute minimum is $-8$ at $x=0.5$ and the absolute maximum is $-\frac{2}{25}$ at $x=5$. The correct answer is choice A.

Find the Absolute Extreme Values of the Function on the Interval