question: 2\nshow this question in the book\nscore: 0 / 1\nscoring method: best answer\n\nan invertible…

question: 2\nshow this question in the book\nscore: 0 / 1\nscoring method: best answer\n\nan invertible function ( f(x) ) is given along with a point that lies on its graph. using theorem 2.7.7, evaluate ( left(f^{-1}\right)(x) ) at the indicated value.\nthe point ( (-9,-749) ) is on the graph of ( f(x)=x^{3}+27 x^{2}+246 x+7 ). find ( left(f^{-1}\right)(-749) ).

question: 2\nshow this question in the book\nscore: 0 / 1\nscoring method: best answer\n\nan invertible function ( f(x) ) is given along with a point that lies on its graph. using theorem 2.7.7, evaluate ( left(f^{-1}\right)(x) ) at the indicated value.\nthe point ( (-9,-749) ) is on the graph of ( f(x)=x^{3}+27 x^{2}+246 x+7 ). find ( left(f^{-1}\right)(-749) ).

Answer

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)): [ \begin{align*} f^\prime(-9)&=3(-9)^{2}+54(-9)+246\ &=3\times81-486 + 246\ &=243-486+246\ &=3 \end{align*} ]

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})