1. Let g(x) is the reflection of f(x) about the line y=x, and f'(x)= x^12/1+x^2, if g(3)=a, then g'(3) =
a) (1+a^2)/a
b) a^12/(1+a^2)
c) (1+a^2)/a^12
d) a/(1+a^2)
Explain, please?
2020-06-09 1:09 pm
回答 (1)
2020-06-09 1:59 pm
"Reflection of f(x) about the line y=x" is another way of saying "inverse function of f(x)". Use the derivative of the inverse formula:
g(x) = f⁻¹(x)
g'(x) = d/dx f⁻¹(x) = 1/f'(f⁻¹(x))
g'(x) = 1 / f'(g(x))
g'(3) = 1 / f'(g(3)) = 1/f'(a)
When I first read the question, I thought there was more than one error in the question, since x^12 doesn't show up in problems very often. As typed, your function simplifies to f'(x) = [(x^12) / 1] + (x^2) = x^12 + x^2; and none of the answers match 1/f'(a) with that interpretation of f.
It looks like you forgot parentheses around (1 + x^2) in the expression for f'(x). That would make (c) the correct answer.
g(x) = f⁻¹(x)
g'(x) = d/dx f⁻¹(x) = 1/f'(f⁻¹(x))
g'(x) = 1 / f'(g(x))
g'(3) = 1 / f'(g(3)) = 1/f'(a)
When I first read the question, I thought there was more than one error in the question, since x^12 doesn't show up in problems very often. As typed, your function simplifies to f'(x) = [(x^12) / 1] + (x^2) = x^12 + x^2; and none of the answers match 1/f'(a) with that interpretation of f.
It looks like you forgot parentheses around (1 + x^2) in the expression for f'(x). That would make (c) the correct answer.
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