Graph f(x) = 2x^4 -4x^2 + 1 and its first two derivatives toge...

Question

Graph f(x) = 2x^4 -4x^2 + 1 and its first two derivatives together. Comment on the behavior of f in relation to the signs and values of f' and f".

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The graph of F of X is equal to 1/4 x 4 minus 2 x 2 + 2. And its first derivative are shown, describe the relationship between the two graphs. Awesome. So it appears when we look at our figure that is provided to us, we have two separate curves. We have our red curve, which denotes Y equals F of X, and we have a green curve, which represents Y equals F of X, which is the first derivative of F of X. So, as we could see, our first curve in red resembles a W shape, whereas our second curve, which is Y equals F of X, which is represented in green, resembles A cubic graph almost means it goes, it starts in the negative 3rd quadrant, and then it goes up, then it curves down, crosses through the origin point, then curve, then it peaks down and then it curves back up again, whereas our red curve does not touch the origin point, rather, it intersects the Y axis at what appears to be at 0.2. Positive to. Awesome, and it looks like our green curve intersects the X axis at 3 separate points and only touches the Y axis at one point. Awesome. So, with that said, now that we have a good visualization of what's happening in this particular prom, first off, we can note that it can be observed that the graph of F is rising whenever F is greater than 0. So once again, it is observed as when we look at our graph, that the function F is rising whenever F is greater than 0. So, for instance, F is rising at 2.0 and at parentheses 2, infinity, specifically positive infinity. So on the same intervals, F is above the X axis 4, F is greater than 0. However, the graph of F, the function of F is falling whenever. F is less than 0. So once again, to quickly recap here, so F, when F is greater than 0, that means the graph of F is rising. So let's make a note of that in blue. So, for this particular case, F is rising, so I have an arrow pointing up. Whereas for our condition F is less than 0, that means that Our function of F is falling, so I have an arrow pointing downwards. Awesome. So focusing on our condition where the graph of our function F is falling whenever F is less than 0. So, for instance, F is falling at parentheses negative infinity -2 in parentheses. And parentheses 02. On the same intervals, so that means that F is below the X axis or F is less than 0. So that means that the graph of F has a local maximum at, so this has a local Max at X equals 0, where the sign of F changes from positive to negative and has a local minimum, so it has a local minima. At X equals -2 and at X equals 2, where the sign of F changes from negative to positive. So that means, without a doubt, our final answer, since we were told ultimately to figure out how to describe the relationship between the two graphs, that's our final answer that we were asked to solve for. So that will mean our final answer without a doubt to help us describe our two graphs will have to be as follows. The graph of F, or function of F is rising whenever F is greater than 0 and falling wherever F is less than 0. The graph of F or function of F has a local max. where the sign of F changes from positive to negative and has a local minimum where the sign of F changes from negative to positive. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graph f(x) = 2x^4 -4x^2 + 1 and its first two derivatives together. Comment on the behavior of f in relation to the signs and values of f' and f".

Key Concepts

Derivative

Second Derivative

Critical Points and Inflection Points

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