Consider the equation f(x) = e^(3x) + e^(−x)?
Consider the equation
f(x) = e^(3x) + e^(−x)
Find the intervals on which f is increasing. (using interval notation.)
Find the interval on which f is decreasing. ( using interval notation.)
Find the local minimum value of f.
Find the interval on which f is concave up. (using interval notation.)
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f'(x) = 3e^(3x) - e^(-x)
f''(x) = 9e^(3x) + e^(-x)
Find critical points:
3e^(3x) - e^(-x) = 0
3e^(4x) - 1 = 0
e^(4x) = 1/3
x = ln(1/3)/4 = -ln(3)/4
The function is increasing on (-ln(3)/4, ∞) and decreasing on (-∞, -ln(3)/4).
f(-ln(3)/4) = 3^(-3/4) + 3^(1/4) = (∜3)/3 + (3∜3)/3 = (4∜3)/3
The function is concave up on (-∞, ∞).