The slope of a curve Y = f(x) is given by dy/dx=(X-1)(x-2)^2(x-3)^3(x-4)^4(x-5)^5.
(a) For what value or values of x does y have a local maximum/minimum? Justify your answer.
I know the maximum/min occurs when dy/dx = 0, which in this case is 1,2,3,4,5 but how do you know which one is max/min?
recall two properties
1. for y = abc
y' = abc' + acb' + bca'
etc for multiple factors
**** However, if any of the a, b, or c are powers such as (c-4)^4 , then that base of x-4 will appear in each of the terms of y'' ***** very important !
2. if y'' > 0 for some x, then that x will produce a minimu, the curve is opening upwards
if y '' < 0 for some x, then that x will produce a maximum, the curve opens downwards
if however y''=0, then there is neither a max or min, the curve "flattens out".
So for your first derivative, we have only one linear factor, the (x-1).
It will appear in all the terms of the second derivative except in the term when it was differentiated.
So when x = 1, you will get zeros for all except
(1)(x-2)^2 (x-3)^3 (x-4)^4 (x-5)^5
finding the value for x = 1 is still easy ...
1(1)(-2)^3(-3)^4 (-4)^5
= (+)(+)(-)(+)(-) = + , notice I don't really care about the actual number value
so x=1 will produce a minimum
all the other factors will appear in each of the terms, so when we sub in x = 2,3, .... we will get 0 for each and thus no max or min for those values.
AHHH ok ok thank you so much!!
To determine whether each value of x corresponds to a local maximum or minimum, we need to analyze the behavior of the slope dy/dx on either side of the value.
For x = 1, the factor (x-1) becomes 0, so dy/dx is undefined at x = 1. Therefore, it is not considered in our analysis for local maximum/minimum.
For x = 2, the factor (x-1) is non-zero. Since (x-1)^2 is positive for any value of x, all other factors (x-2)^2, (x-3)^3, (x-4)^4, and (x-5)^5 are also positive. Therefore, dy/dx is positive for x close to 2. This suggests that y is increasing as x approaches 2. Therefore, there is no local maximum/minimum at x = 2.
Similarly, we can analyze the behavior for x = 3, 4, and 5. For all three values, we can observe that dy/dx is negative for x close to each value. This suggests that y is decreasing as x approaches 3, 4, and 5. Therefore, there are no local maximum/minimum at x = 3, 4, or 5.
In summary, based on the analysis of the factors and the behavior of dy/dx on either side of the values, we conclude that the given curve Y = f(x) does not have any local maximum or minimum.