(5 points) Letf(r) = ~x" 9x' + 4x ~ 6. Find the open intervals on which f is concave up (down}; Then determine the x-coordinates of all inflection points ...

Find the intervals on which the graph off is concave upward, the intervals on which the graph off is concave downward, and the $x$ coordinates of the inflection points. $$ f(x)=x^{4}+6 x $$

Mhm. Yeah. In this problem we want to find where the graph is concave versus down as well as the inflection points of the function after max equals X to the fourth plus six X. As is noted here, we're going to be using the second derivative to an Alaskan cavity Inflection points. Following the five steps outlined below the versions that one we find the second derivative of F F prime is four X cubed plus six. Therefore eligible prime is 12 X squared. And step two. We find the partition points where the function is undefined in the denominator or equals zero. Unless we have 12 X squared equals zero, which gives X equals zero as a partition point. That's on our sign chart, we have to evaluate the sign after the prime, left and right of zero, From negative infinity to zero after the crime is positive From zero infinity. Ethical crime is also positive. Thus, in step floor, we conclude that the function is concave up on zero, Union, infinity. Concave down nowhere. In Step five, we conclude that inflection points, since they only occur where there are changes in lung cavity, none must exist for this function. There are no inflection points.

Mhm. Mhm. We want to find where the graph is concave up versus down as well as the inflection points. As the note here mentions, we're going to be using the second derivative to analyze can cavity inflection points whereby we follow the five steps listed on the left to do so. So instead, when we find F double prime, F prime is four X cubed plus 12 X. Therefore F double primer simply 12 X squared plus 12. The partition points are where after double prime equals zero. So 12 X squared plus 12 equals zero, gives x squared equals negative one. This has no solutions. So there are no partition points. That's on our science art. We simply evaluate the sign it after the prime on all X. Or negative financial affinity because after the problem is positive for negative infinity to infinity. We are able to conclude in step four that the function is concave up for all X. On that infinity to infinity. The function is concave down nowhere. Finally, inflection points occur where there's changes in con cavity since there is no such changes, there are no inflection points or the function

We want to find where the graph is. Conch you up versus down as well as the inflection point. For the function after backs equals x q minus four X squared plus five x minus two. This question is chest. They are understanding of how to use a secondary to analyze con cavity or inflection points. Obviously shown by the outline here. We're going to use the five steps listed on this template to solve In step one. We use double prime, the second derivative. We have to find it. That is f prime equals three experiments eight plus five which gives five double prime equals 66 minutes eight. We just find the partition points. This is where the problem is equal to zero or undefined. Thus we have six X minus eight equals zero giving eight equals Or another. X equals 4/3. That's on our side. We evaluate the sign of a little time left and right up 4/3. This gives from negativity for 3rd septal primary negative from 430 to infinity after the prime is positive. Therefore, took on Cavity is concave up from 4 30 to Infinity. Concave down a negative affinity to 4/3. Inflection points occur where there's changing from cavity, so we conclude that X equals four thirds is our inflection point.

In this problem to find the places where the graph con caves up or down. We will find the information by making the sign chart for F double dash X. Also the point of X where F double dash X will change its sign will give us the infection we have If x equals two X cubed minus four, it's square Plus five X -2. Supposed to find the derivative, this is three X square minus. It takes last five. The next generator, This is 66 minus eight. Now we'll find partition number to help us make the sign chart for f double checks, partition number. It occurs when F double dash x is either zero or it does not exist. We see that it exists six X men as it exists for all values of X. So we'll just equated to zero from where we get X sequence to fought by three. Now we will make the sign chart for this Says four x 3 have double dash x. Take test value of zero and 2 F double last zero Is -8 which is negative. And F double lash too. Mhm. 12 minutes it which is four which is positive. Means that this reason after backlash access positive. Yeah. And then this reason it is negative at four x 3. It is president. We see that double large extend this sign. Aren't X X equals to four by three. So the inflection point As x equals two. Four by three. The corresponding why value is equal to affect four x 3? Which can be calculated. It becomes four by three. All square whole cube minus four. Came four by three squared plus five times four by three. Right, This is to comes us minus 0.07. Well, we'll talk about where the fx graph of effects will can't give up and can't get down very of this. F double check is negative. Did the draft will be come kids town And in this region of this on Kings. Uh huh. So from minus infinity to four x 3 deaths. One came down and from for about 3 to Infinite. It's compared to that.