F(X)=e^(4X)+e^(-x) a. Find the intervals on which f is increasing and decreasing. b. Find the local minimum value of f. c.  Find the interval on which f is concave up.

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Given `f(x)=e^(4x)+e^(-x)` :

(1) `f'(x)=4e^(4x)-e^(-x)`

(2) `f''(x)=16e^(4x)+e^(-x)`

(3) A function is increasing if the first derivative is positive, and decreasing if the first derivative is negative.

`4e^(4x)-e^(-x)>0`

`4e^(4x)>e^(-x)`

`ln[4e^(4x)]>ln[e^(-x)]`

`ln4+lne^(4x)>lne^(-x)`

`ln4+4x > -x`

`x>(-ln4)/5~~-.2772588722`

(The entire section contains 185 words.)

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