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# Justification using first derivative

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.10 (EK)
,
FUN‑4.A.11 (EK)

## Video transcript

the differentiable function f and its derivative F prime are graphed so let's see we see the graph of y is equal to f of X here in the blue and then F prime we see in this brownish orangish color right over here what is an appropriate calculus based justification for the fact that F is decreasing when X is greater than 3 so we can see that that actually is indeed the case when X is greater than 3 we see that our function is indeed decreasing as x increases the Y value the value of our function decreases so a calculus based justification without even looking at the choices well I could look at the derivative and we're going to be decreasing if the slope of the tangent line is negative which means that the derivative is negative and we can see that for X is greater than 3 the derivative is less than zero so my justification I haven't even looked at these choices yet I would say for X is greater than 3 f prime of X is less than zero that would be my justification not even looking at these choices now let's look at the choices F prime is decreasing when X is greater than 3 now this isn't right what we care about is whether f prime is positive or negative if f prime is negative if it's less than zero then the function itself is decreasing the slope of the tangent line will be negative you could F prime could be positive while decreasing for example F prime could be doing something like this and even though f prime would be decreasing in this situation the the actual value of the derivative would be positive which means the function would be increasing in that scenario so I would rule this one out for values of X larger than 3 as X values increase the values of f of X decrease now that is actually true this is actually the definition that F is decreasing as X values increase the values of f of X decrease but this is not a calculus based justification so I am going to rule this one out as well F prime is negative when X is greater than three well that's exactly what I wrote up here if F prime is negative then that means that our slope of the tangent line of our original function f is going to be downward sloping or that our function is decreasing so this one is looking good and this one right over here says f prime of zero is equal to negative three so they're just pointing out this point this isn't even relative to the interval that we care about or this isn't even relative min when X is greater than three so we definitely want to rule that one out let's do one more of these so here we're told the differentiable function G and it's derivative G prime are graphed so once again G is in this bluish color and then G prime its derivative is in this orange color what is an appropriate calculus based justification for the fact that G has a relative minimum point at X is equal to negative three and we could see it here when X is equal to negative 3 it looks like G is equal to negative 6 and it looks like a relative minimum point there so what's the best justification so once again without even looking at the choices I would say a good justification is before we get to x equals negative 3 before we get to x equals negative 3 our derivative and there's a calculus based justification before we go to x equals negative 3 our derivative is negative is negative and after x equals negative 3 our derivative is positive that would be my justification because if our derivative is positive before that valid if our derivative is negative before that value that means that we are downward sloping before that value and if it's positive after that value that means we're upward sloping after that which is a good justification that we are at a relative minimum point right over there so let's see the point where x equals negative 3 is the lowest point on the graph of G in its surrounding interval that is true but that's not a calculus based justification you wouldn't even have to look at the derivative to make that statement so let's rule that one out G prime has a relative maximum at 0 comma 3 at 0 zero comma three it actually does not G prime G prime yes G prime actually does have a relative maximum at 0 comma 3 but that doesn't tell us anything about whether we're at a at a relative minimum point at x equals negative 3 so I would rule that out G prime of negative 3 is equal to 0 so G prime of negative 3 is equal to 0 so that tells us that the slope of the tangent line of our function is going to be 0 right over there but that by itself is not enough to say that we are the relative minimum point for example I could at be at a point that does something like this where the slope of my tangent line is 0 and then it keeps increasing again or it does something like this and it keeps decreasing so even though you're at a point where the slope of your tangent line is 0 it doesn't mean you're at a relative minimum point so I would rule that out G prime crosses the x-axis from below it to above it at x equals negative 3 G prime crosses the x-axis from below it to above it yep and that's the argument that I made that we're going from below the x-axis so G prime goes from being negative to positive which means the slope of the tangent lines of our points as we approach x equals negative 3 go from being downward sloping to upward sloping which is an indication that we are at a relative minimum point
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