## 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) ?

### To find the expression for g'(2), we need to first find the inverse function g(x) of f(x)

To find the expression for g'(2), we need to first find the inverse function g(x) of f(x).

Given: f(x) = x^3 + 2x^2 + 4x + 5

To find the inverse function, we need to swap the positions of x and y and solve for y.

x = y^3 + 2y^2 + 4y + 5

Rearranging the equation:

y^3 + 2y^2 + 4y + 5 – x = 0

Next, we solve for y in terms of x, which can be done by using various methods such as factoring, synthetic division, or numerical methods.

However, in this case, since the function f(x) is strictly increasing, we know that its inverse function g(x) will be strictly increasing as well. This means that g'(x), the derivative of g(x), will always exist and be positive.

Using this information, we can proceed to find g'(2) by taking the derivative of the inverse function g(x) at x = 2.

g'(2) = f'(g(2))^-1

Substituting x = 2 into the equation for f(x), we can find g(2):

2 = 2^3 + 2(2)^2 + 4(2) + 5

2 = 8 + 8 + 8 + 5

2 = 29

So, g(2) = 29.

Now, we need to find f'(g(2)):

f'(x) = 3x^2 + 4x + 4

Substituting x = 29 into the equation for f'(x), we can find f'(g(2)):

f'(29) = 3(29)^2 + 4(29) + 4

f'(29) = 3(841) + 116 + 4

f'(29) = 2523 + 116 + 4

f'(29) = 2643

Finally, we take the inverse of f'(29) to find g'(2):

g'(2) = (f'(29))^-1

g'(2) = 1/2643

So, the correct expression for g'(2) is 1/2643.

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