Determine the largest open intervals of the domain over which eac...

Question

Determine the largest open intervals of the domain over which each function is (b) decreasing. See Example 8.

Accepted Answer

Welcome back. I am so glad you're here. We're told for the given function find the domain over which the function is decreasing. Then we have a graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin, the domain for what's drawn for our X axis is from negative eight to positive eight. And that's marked off in increments of one half. And the range for what's drawn for our Y axis is from negative 50 to positive 300. And that's marked off in increments of 10. On top of our graph, we have a function line and it looks kind of like a curvy W in the second quadrant, the function line is heading up toward positive infinity for the Y values and more slowly toward negative infinity for our X values. Then it's heading down to the point negative 312 where it turns around and heads up until it gets to the 0.0 93 when it heads down again until it gets to the 0.3 12 where it heads up toward positive infinity for both the X and Y values, but more quickly toward positive infinity for the Y values than the X values. Our answer choices are answer choice A from negative infinity to negative three answer choice B from 0 to 3 answer choice C from negative infinity to negative three in union with 0 to 3 and answer choice D none. All right. So what are we being asked here? We're asked to find the domain. Those are our X values over which the function is decreasing. And what that means is where are Y values are decreasing or going down as we move from left to write for our ex values. So we begin on the left and we see what our why values do as we move to the right. So on the left here, that's negative infinity. Our function is way up toward positive infinity for the Y values. And as we head to the right, our Y values are going down, they go all the way down to 12 where the X value is negative three before they turn around. So part of the domain where this is decreasing is from the X values of negative infinity to negative three. But that's not all we see at the X value of negative three. It's our Y values are heading up from 12 all the way up to 93 when our X value is zero. However, there it turns around and starts heading down again. The function is heading down from X values of zero to three when the Y values go from 93 to 12. So that's another section. So we say that we have our first domain from negative infinity to negative three. And that's in union with the next section which is our X values from 0 to 3. And then at the X value of three, our Y values start going up again. They go from 12 all the way up to positive infinity as our X values head toward positive infinity. So we've got our domain. It's from negative three or excuse me, not negative three, negative infinity to negative three in union with 0 to 3. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Determine the largest open intervals of the domain over which each function is (b) decreasing. See Example 8.

Key Concepts

Function Behavior

First Derivative Test

Open Intervals

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