answer the following true or false: if ( g(x) ) is the inverse of a differentiable function ( f(x) ) with…

answer the following true or false: if ( g(x) ) is the inverse of a differentiable function ( f(x) ) with derivative ( f^{prime}(x)=6+sin left(x^{2}\right) ), then ( g^{prime}(0)=\frac{1}{6} ). true false

answer the following true or false: if ( g(x) ) is the inverse of a differentiable function ( f(x) ) with derivative ( f^{prime}(x)=6+sin left(x^{2}\right) ), then ( g^{prime}(0)=\frac{1}{6} ). true false

Answer

Explanation:

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

If (g(x)) is the inverse of (f(x)), then (g^{\prime}(y)=\frac{1}{f^{\prime}(x)}) where (y = f(x)).

Step2: Find (x) such that (f(x)=0)

We need to find (x) for which (f(x) = 0). Let's assume (x = 0). Then (f(0)=\int_{0}^{0}(6+\sin(t^{2}))dt=0) (by the fundamental theorem of calculus, (\int_{a}^{a}h(t)dt = 0) for any function (h(t))).

Step3: Calculate (f^{\prime}(x)) at (x = 0)

Given (f^{\prime}(x)=6+\sin(x^{2})), when (x = 0), (f^{\prime}(0)=6+\sin(0)=6)

Step4: Use the formula for (g^{\prime}(y))

Since (y = f(x)=0) when (x = 0), and (g^{\prime}(y)=\frac{1}{f^{\prime}(x)}), substituting (y = 0) and (x = 0) gives (g^{\prime}(0)=\frac{1}{f^{\prime}(0)})

Since (f^{\prime}(0)=6), then (g^{\prime}(0)=\frac{1}{6})

Answer:

True