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 find the derivative of the inverse function g at x = 2
To find the expression for g'(2), we need to find the derivative of the inverse function g at x = 2.
First, let’s find the inverse function g(x) of f(x). To do this, we swap the roles of x and y in the equation f(x) = x^3 + 2x^2 + 4x + 5 and solve for x:
x = y^3 + 2y^2 + 4y + 5
Now, let’s solve this equation for y. This will give us the inverse function g(x):
y^3 + 2y^2 + 4y + 5 – x = 0
There is no simple algebraic solution to find the inverse function, so we will have to use numerical methods or approximate the inverse function.
However, we can still calculate the derivative of g(x) at a specific point using implicit differentiation. Let’s find g'(y) first:
d/dx (x^3 + 2x^2 + 4x + 5 – y) = 0
3x^2 + 4x + 4 – dy/dx = 0
Now, let’s substitute y = g(x) into the equation:
3x^2 + 4x + 4 – g'(x) = 0
Since we are looking for g'(2), we substitute x = 2 into the equation:
3(2)^2 + 4(2) + 4 – g'(2) = 0
Simplifying this equation:
12 + 8 + 4 – g'(2) = 0
24 – g'(2) = 0
g'(2) = 24
Therefore, the correct expression for g'(2) is 24.
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