Let f be the increasing function defined by f(x)=x3+2×2+4x+5, where f(−1)=2. If g is the inverse function of f, which of the following is a correct expression for g′(2) ?
Correct. This can be confirmed using the chain rule and the definition of an inverse function. Since f(g(x))=xf(g(x))=x, it follows that ddxf(g(x))=f′(g(x))g′(x)=ddx(x)=1⇒g′(x)=1f′(g(x))ddxf(g(x))=f′(g(x))g′(x)=ddx(x)=1⇒g′(x)=1f′(g(x)). Therefore, g′(2)=1f′(g(2))=1f′(−1)g′(2)=1f′(g(2))=1f′(−1).
First, we can find the value of f(0) by substituting x=0 into the given function:
f(0) = 0^3 + 2(0)^2 + 4(0) + 5 = 5
Next, we can use the inverse function theorem to find g'(2), which states that if f(x) has an inverse function g(x), then g'(x) = 1/f'(g(x)).
We can find f'(x) by taking the derivative of the given function:
f'(x) = 3x^2 + 4x + 4
Since g is the inverse function of f, g(2) must equal 0 (since f(0) = 2 and f(2) > f(0) since f is increasing).
Therefore,
g'(2) = 1/f'(g(2))
= 1/f'(0)
= 1/(3(0)^2 + 4(0) + 4)
= 1/4
Thus, the correct expression for g'(2) is 1/4.
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