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Part C find all values of X on the interval negative 4 is less than X is less than 3 for which the graph of G has a point of inflection give a reason for your answer so an inflection point an inflection point is a point where the sign of the second derivative sign of second derivative changes changes so if you take the second derivative at that point or as we go close to that point or as we cross that point it goes from positive to negative or negative to positive and to think about that visually you can think of some examples so if you have a curve that looks something like this if you have a curve that looks something like this you'll notice that over here the slope is negative but it's increasing it's getting less negative let's let negative then it goes to 0 then it keeps increasing slope is increasing increasing all the way to there and then it starts getting less positive so it starts decreasing so it's increasing the slope is increasing over at this point right over here so even though the slope is negative it's getting less negative over here so it's increasing and then the slope keeps increasing getting getting more and more positive up to about this point and then the slope is positive but then it becomes less positive so the slope begins decreasing after that so then the slope begins decreasing after that so this right over here is a point of inflection the the the slope has gone from increasing to decreasing and the other thing happened if the slope went from decreasing to increasing that would also be a point of inflection so if this was maybe some type of a trigonometric curve then you might see something like this and so this also this also would be a point of inflection but for this you know our G of X is kind of hard to visualize the way they've defined it right over here so the best way to think about it is just figure out where its second derivative has a sign change and to think about that we have to find its second derivative so let's write G of x over here we know G of X is equal to 2x plus the definite integral from 0 to X of F of T DT we've already taken its derivative but we'll do it again G prime of X is equal to 2 plus fundamental theorem of calculus the derivative of this right over here is just f of X and if we have the second derivative of G G prime prime of X this is equal to the derivative of 2 is just 0 and the derivative of f of X is f prime of X so asking where this has a sign change asking where our second derivative has a sign change is equivalent to asking where does the first derivative of f have a sign change sign change and asking where the first derivative of f has a sign change is equivalent to saying where does the slope of F have a sign change you could view this as a the slope the slope or the instantaneous slope of F so we want to know where that when the slope of F has a sign change so let's think about it over here the slope is positive it's going up it's up it's getting it's increasing but it's positive and that's what we care about so let's write it or other than green so the slope is positive this entire time it's increasing its increasing it's positive it's getting less positive now it's starting to decrease but the slope is still positive the slope is still positive all the way until we get right over there it seems like it gets pretty close to 0 and then the slope gets negative and then right over here the slope is negative the slope is negative the slope is negative right over here so this is interesting because because even though F is actually not differentiable right here so f is not differentiable at that point right over there and you can see because the slope goes pretty close to 0 and then it just jumps to negative 3 so you have a discontinuity you have a discontinuity of the derivative right over there but we do have a sign change we go from having a positive a positive slope on this part of the curve to having a negative slope over this part of the curve so we experienced a sign change right over here at X is equal to 0 a sign change in the first derivative of F which which is the same thing as saying a sign change in the second derivative of G and a signs change in the second derivative of G tells us that when X is equal to 0 that when X is equal to 0 we have a point the graph of G has a point of inflection
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