Answer:
6.93
Step-by-step explanation:
f(x) = 15 / (1 + 4e^(-0.2x))
f(x) = 15 (1 + 4e^(-0.2x))^-1
Taking first derivative:
f'(x) = -15 (1 + 4e^(-0.2x))^-2 (-0.8e^(-0.2x))
f'(x) = 12 (1 + 4e^(-0.2x))^-2 e^(-0.2x)
f'(x) = 12 (1 + 4e^(-0.2x))^-2 (e^(0.1x))^-2
f'(x) = 12 (e^(0.1x) + 4e^(-0.1x))^-2
Taking second derivative:
f"(x) = -24 (e^(0.1x) + 4e^(-0.1x))^-3 (0.1e^(0.1x) − 0.4e^(-0.1x))
Set to 0 and solve:
0 = -24 (e^(0.1x) + 4e^(-0.1x))^-3 (0.1e^(0.1x) − 0.4e^(-0.1x))
0 = 0.1e^(0.1x) − 0.4e^(-0.1x)
0.1e^(0.1x) = 0.4e^(-0.1x)
e^(0.1x) = 4e^(-0.1x)
e^(0.2x) = 4
0.2x = ln 4
x = 5 ln 4
x ≈ 6.93
Graph: desmos.com/calculator/zwf4afzmav
