f(x)=8x-2*ex
Find the derivative.
f'(x)=8-2*ex
Set the derivative equal to 0.
Solve for x.
x = ln4
After finding the point that makes the derivative f'(x)=8-2*ex equal to 0 or undefined, the interval to check where f(x)=8x-2*ex is increasing and where it is decreasing is ( − ∞ , ln4 ) ∪ ( ln4 , ∞ ).
( − ∞ , ln4 ) ∪ ( ln4, ∞ )
Substitute a value from the interval (-∞,4) into the derivative to determine if the function is increasing or decreasing.
Increasing on (-∞,ln4) since f ' ( x ) > 0
Substitute a value from the interval (4,∞) into the derivative to determine if the function is increasing or decreasing.
Decreasing on ( ln4, ∞ ) since f ' ( x ) < 0
List the intervals on which the function is increasing and decreasing.
Increasing on: ( − ∞ , ln4)
Decreasing on: ( ln4, ∞ )
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