Sketch the graph of a continuous function ƒ on [0, 4] satisfyi...

Question

Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.

ƒ' (x) = 0 for x = 1 and 2; ƒ has an absolute maximum at x = 4; ƒ has an absolute minimum at x= 0; and ƒ has a local minimum at x = 2.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says determine the properties that describe the graph of F of X on the closed interval from 1 to 6.5. Below the problem, we're given a graph that has our function F of X plotted on it. Our function begins at the point of 1.17 and ends at the point of 6.5.0. Our function is concave up on the interval from 1 to 4, and it's concave down on the interval from 4 to 6.5. Now this problem wants us to determine the properties that describe the graph of our function. And so we want to start off by looking at our first derivative and determine the points when it's equal to 0. So F of X is equal to 0. And here we're looking for the points where our slope is equal to 0, and if we look at our graph, this occurs at X equal to 3 and X equal to 5. So now we're gonna look for local minima and maxima. And we see that we have a local. Minimum. When x is equal to 3. And that's because for points just to the left of X equal to 3, we see that our function F of X is decreasing, and to points just to the right of X equals 3, we see that our function is increasing. So that means we have a local minimum. We also have a local maximum. At X equal to 5. And that's because at X equal to 5, just to the left of that, or if we look at points just to the left of that, we see that our function is increasing, and to the points just to the right of X equal to 5, we see that our function is decreasing. That gives us a local maximum. So now we need to actually identify our absolute minimum and maximum. So, our absolute maximum. is occurring at X equal to 1. Cause at X1, that is the highest point on our graph at Y equal to 17. And the absolute minimum. Which is going to be the lowest point on our graph. Occurs at X equal to 6.5, since the Y value at that point is 0, and that is the lowest point. So now all we need to do is identify the intervals over which our function is increasing or decreasing. And we see that our function is decreasing, that means that F of X is less than 0. On the open interval from 1 to 3. And the open interval from 5 to 6.5. And our function is increasing, meaning that F of X. is greater than 0. On the open interval from 3 to 5. So, these are all the properties that we needed to determine to describe this graph. So, we started off by finding the points which our first derivative was equal to 0. We didn't categorize those points as either a local minimum or a local maximum. We then found the absolute maximum and the absolute minimum for our function. And then we determined the intervals over which our function was decreasing and over which it was increasing. But this here is the answer to this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.

ƒ' (x) = 0 for x = 1 and 2; ƒ has an absolute maximum at x = 4; ƒ has an absolute minimum at x= 0; and ƒ has a local minimum at x = 2.

Key Concepts

Derivative and Critical Points

Absolute and Local Extrema

Continuity of Functions

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