SOLUTION: Let f (x) = e^x - e^4x Find all extreme values (if any) of f on the interval [0, 1]. Determine at which numbers in the interval these values occur. Thank you

Algebra ->  Equations -> SOLUTION: Let f (x) = e^x - e^4x Find all extreme values (if any) of f on the interval [0, 1]. Determine at which numbers in the interval these values occur. Thank you      Log On


   

Question 996073: Let f (x) = e^x - e^4x
Find all extreme values (if any) of f on the interval [0, 1]. Determine at which numbers in the interval these values occur.
Thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1) Apply the derivative (use chain rule)
f(x) = e^x - e^(4x)
f ' (x) = e^x - 4*e^(4x)


Step 2) Solve f ' (x) = 0 for x to find the critical values
f ' (x) = e^x - 4*e^(4x)
0 = e^x - 4*e^(4x)
e^x = 4*e^(4x)
e^x/[e^(4x)] = 4
e^(x-4x) = 4
e^(-3x) = 4
-3x = ln(4)
x = (-1/3)*ln(4)
x = -0.462098 ... this is approximate

Since the x value of -0.462098 is NOT in the interval [0,1], this means that there are NO extrema on f(x) in the interval [0,1]