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PerfectMagZ
04.12.2019 •
Mathematics
Which description best explains the domain of (gºf)(x)?
the elements in the domain of f(x) for which g(f(x)) is defined
the elements in the domain of f(x) for which g({(x)) is not zero
the elements in the domain of g(x) for which g{{(x)) is defined
the elements in the domain of g(x) for which tr(x)) is not zero
o
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Ответ:
A. The elements in the domain of f(x) for which g(f(x)) is defined
I just took the test myself. (gºf)(x) can be rewritten as g(f(x)) and the interior function, which is f(x) in this case, is the domain, where x is defined.
Ответ:
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