Secant lines: challenging problem 1

consider the graph of the function f of X that passes through three points as shown so these are the three points in this curve and blue is f of X identify which of the statements are true so they give us these statements let's see this first one says F of negative a is less than one minus F of negative a over a so this seems like some type of bizarre statement how do we able to figure out whether this is true from this right over here so let's just go piece by piece and see if something starts to make sense so F of negative a where do we see that here well this is F of negative a this is the point this is the point x equals negative a so this is negative a and this is y is equal to f of negative a so this is F of negative a right over here and what we can what we know about F of negative a based on looking at this graph is that F of negative a is between 0 & 1 so we could write that 0 is less than F of negative a which is less than 1 so that's all I can deduce about F of negative a right from the get-go now let's look at this crazy statement 1 minus 1 minus F of negative a over a what is this well let's think about what happens if we take the secant line if we're trying to find the slope of the secant line between this point and this point if we want to find the average rate of change between the point negative a f of negative a and the point 0 1 if this is our end point our change in Y our change in Y is going to be 1 minus F of negative a so 1 minus F of negative a is equal to our change in Y our change in Y and our change in X our change in X going from negative a to 0 so change in X is going to be equal to 0 minus negative a which is equal to positive a so this right over here is essentially our change in x over our change in Y from this point to this point it is our average it is our average rate of change from this point to this point so it is our average rate of change or you could say it's the slope of the secant line so the secant line would look something like this something something like this slope of the secant line so this right over here is slope of secant secant line between from from negative a f of negative a to zero comma one so just looking at this diagram right over here what do we know about this slope and in particularly make any statements about that slope relative to say zero or one or anything like that well let's think about what is what a line of slope one would look like well a line of slope one especially one that went through this point right over here would look something like would look something like this a line of slope one would look something like this so this line right over here that I've just drawn that goes from negative 1 comma 0 to 0 1 this has slope 1 so this slope is equal to 1 so if this green line has a slope of 1 does this blue line have a slope as clearly has a different slope is that slope is this blue line steeper or less deep than the green line well it's pretty clear that this this secant line is steeper than the green line it's increasing faster so it's going to have a higher slope so this looking at it from this diagram this blue line has a slope higher than 1 or the slope of the secant line from negative a f of negative a to 0 comma 1 that is going to be greater than 1 so this thing right over here is greater than 1 so we're able to do is this thing right over here is less than 1 this thing over here is greater than 1 so this thing is less than that thing so this must be true now let's look at this one we're comparing the slope of the secant line we're comparing the slope of the secant line that we just looked at so this is the same value right over here so we're comparing this slope right over here - what's this F of a minus 1 over a well this is the slope of this secant line this is the slope of this secant line that I'm drawing in this let me do it more contrast let me do it in orange that is the slope of this of this secant line the slope of this secant line so which one has a higher slope well it's pretty clear that the blue secant line has a higher slope than this orange secant line but here it's saying that the blues slope is lower than the orange so this is not going to be true so this is not true then finally let's look at this over here F of a minus F of negative a over 2a so they're now complete there now is this is the slope let me draw this so this right over here this is the slope of the secant line between this point and this point right over here our change in Y is f of a minus F of negative a our change in X is a minus negative a which is two ways so this is the secant line right over here so let me draw it so this secant line the secant line right over here so they're comparing that slope to two this slope F of a minus one is our change in Y over a is our change in X so we're comparing it to that one right over there and you could immediately eyeball this kind of brownish maroon it I guess it's kind of a brown color this this this secant line that goes all the way from here to here it's clearly steeper than this one right over here and we know that the average rate of change from here to here is going to be higher than the average rate of change from here here because at least from negative a to zero we were increasing at a much faster rate and then we slowed down to this rate so the average over the entire interval is definitely going to be more than this then the what we get from zero to a so this one is also not true this has a higher we actually know that this is false if this both of these would have been true if we swapped these signs around if this was a greater than sign if this is a greater than sign so this is the only one that applies