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an invertible function ( f(x) ) is given along with a point that lies on its graph. using theorem 2.7.7, evaluate ( left(f^{-1}
ight)(x) ) at the indicated value.
the point ( (-9,-749) ) is on the graph of ( f(x)=x^{3}+27 x^{2}+246 x+7 ). find ( left(f^{-1}
ight)(-749) ).

Explanation:

Step1: Recall the formula for the derivative of the inverse function

If \(y = f(x)\) is invertible and differentiable, then \((f^{-1})^\prime(y)=\frac{1}{f^\prime(x)}\), where \(y = f(x)\). Here, \(y=-749\) and \(x = - 9\) (since the point \((-9,-749)\) lies on the graph of \(y = f(x)\)).

Step2: Find the derivative of \(f(x)\)

First, find \(f^\prime(x)\) using the power rule. If \(f(x)=x^{3}+27x^{2}+246x + 7\), then \(f^\prime(x)=3x^{2}+54x + 246\).

Step3: Evaluate \(f^\prime(x)\) at \(x=-9\)

Substitute \(x = - 9\) into \(f^\prime(x)\):

$$
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$$

Step4: Use the formula for \((f^{-1})^\prime(y)\)

Since \((f^{-1})^\prime(-749)=\frac{1}{f^\prime(-9)}\) and \(f^\prime(-9) = 3\).

Answer:

\(\frac{1}{3}\)