Find the absolute maximum and minimum values of the function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. f(x) = -x - 4, -4 <= x <= 1

Solutions

Verified$f(x)=-x-4, -4\leq x\leq1$

Graph of function

Since $f(x) =-x - 4$ is continuous on the closed interval $[-4, 1]$, by Theorem 1 (i.e., the Extreme Value Theorem), the absolute extremum values must exist. They can occur at either endpoint of the interval or at a critical point of $f$ that is inside the interval.

Now let's find critical points of $f$ (if any):

$f' = -1$

Since $f'$ can't be 0 or undefined, there are no critical points of $f$. In other words, the absolute extremum values must occur at the endpoints of the interval.

At the left endpoint $x = -4$:

$f(-4) = -(-4) - 4 = 0$

At the right endpoint $x = 1$:

$f(1) = -1 - 4 = -5$

Since $0$ is the larger of the two values, $0$ is the absolute maximum. Likewise, since $-5$ is the smaller of the two values, $-5$ is the absolute minimum. (See graph on the right.)

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