Find the absolute maximum and minimum values of each function on the given interval. f(x)=x(e^-x)
df/dx = - x e^-x + e^-x
= 0 for max or min within interval
BUT you did not say the interval
in general
e^-x = x e^x
1 = x is one place where the slope is zero
is the second derivative + or - ?
d^2f/dx^2 = -x(-e^-x) -e^-x - e^-x
= xe^-x -2 e^-x)
= (1/e^x)(x-2)
when x = 1
= (1/e) (-2) which is - so that is a max
now I do not know your interval so you better check the ends., I suspect the linmit of x/e^x ---> 0 as x ---> infinity so that would end up being a minimum
Oh and by the way when x = 0 we have 0/1
To find the absolute maximum and minimum values of a function on a given interval, you need to follow these steps:
Step 1: Determine the critical points of the function within the interval by finding the values of x that make the derivative equal to zero or undefined.
Step 2: Evaluate the function at these critical points, along with the endpoints of the interval.
Step 3: Compare the values obtained in step 2 to determine the absolute maximum and minimum.
Now, let's apply these steps to the function f(x) = x(e^-x) on a specific interval. Please provide the interval for which you want to find the absolute maximum and minimum values.