Find the open​ interval(s) on which the function is increasing and decreasing, and Identify the​ function's local and absolute extreme​ values, if​ any, saying where they occur.

Question

f(x)= e^15x + e^-x

Answer 1

f(x) = e^15x + e^-x

Let u = 15x and v = -x. Apply the chain rule:

df(x)/dx = de^u/du · du/dx + de^v/dv · dv/dx

df(x)/dx = 15e^15x - e^-x

To find the extreme points, set the derivative to zero and solve for x:

0 = 15e^15x - e^-x

e^-x = 15e^15x

1 = 15e^16x

1/15 = e^16x

ln(1/15) = 16x

ln(1/15)/16 = x ≈ -0.17

There is only on extreme point. To determine if it's a maximum or minimum, take the second derivative and plug in x = -0.17. If the second derivative is positive at that point, then it's a minimum and the function is decreasing from (-∞,-0.17) and increasing from (-0.17,∞). If the second derivative is negative, then the extreme point is a maximum and the function is increasing from (-∞,-0.17) and decreasing from (-0.17,∞).