Consider the function f(x) = 7 xe^{-7.5 x}, 0<_x<_2.



This function has an absolute minimum value equal to:

which is attained at x=

First step is usually to find the derivative:

f = 7xe^(-7.5x)
f' = 7e^(-7.5x) + 7xe^(-7.5x)(-7.5)
= 7e^(-7.5x) (1-7.5x)

since e^(ax) is never zero, we just need to see when (1-7.5x) = 0.

I think you can take it from there.

The graph is at

http://www.wolframalpha.com/input/?i=7xe^%28-7.5x%29+for+-.1+%3C+x+%3C+.5

Be careful. It's almost a trick question.

To find the absolute minimum value of a function, we need to find the critical points and evaluate the function at those points.

To find the critical points, we need to find where the derivative of the function is equal to zero or undefined.

First, let's find the derivative of f(x):

f'(x) = 7 e^(-7.5 x) - 7.5x e^(-7.5 x)

Next, we set the derivative equal to zero and solve for x:

7 e^(-7.5 x) - 7.5x e^(-7.5 x) = 0

Factor out the common term e^(-7.5 x):

e^(-7.5 x) (7 - 7.5x) = 0

Setting each factor equal to zero:

e^(-7.5 x) = 0 (not possible)

7 - 7.5x = 0

Solving for x:

x = 7/7.5 = 14/15

Now that we have found the critical point x = 14/15, we can evaluate the function at that point:

f(14/15) = 7 * (14/15) * e^(-(7.5 * 14/15))

Using a calculator or computer software, we can evaluate this expression to find the absolute minimum value of f(x) at x = 14/15.