let (g(x)) be the inverse of (f(x)=x^{3}+2x + 2). without finding a formula for (g(x)), calculate (g(5))…

Explanation:

Step1: Recall inverse - function property

If (g(x)) is the inverse of (f(x)), then (f(g(x))=x) and (g(f(x)) = x). We want to find (g(5)), so we need to find (x) such that (f(x)=5). That is, we solve the equation (x^{3}+2x + 2=5), or (x^{3}+2x - 3=0).

Step2: Factor the equation

We can factor (x^{3}+2x - 3) as ((x - 1)(x^{2}+x + 3)=0). The quadratic factor (x^{2}+x + 3) has discriminant (\Delta=1^{2}-4\times3=- 11<0), so its roots are complex. The real - root of (x^{3}+2x - 3 = 0) is (x = 1). So (g(5)=1).

Step3: Recall the formula for the derivative of an inverse function

The formula for the derivative of an inverse function is (g^{\prime}(y)=\frac{1}{f^{\prime}(g(y))}). We want to find (g^{\prime}(5)), so we first find the derivative of (f(x)). Using the power rule, (f^{\prime}(x)=3x^{2}+2).

Step4: Evaluate (f^{\prime}(g(5)))

Since (g(5)=1), we evaluate (f^{\prime}(1)). Substitute (x = 1) into (f^{\prime}(x)): (f^{\prime}(1)=3\times1^{2}+2=5).

Step5: Calculate (g^{\prime}(5))

Using the formula (g^{\prime}(5)=\frac{1}{f^{\prime}(g(5))}), and since (f^{\prime}(g(5))=f^{\prime}(1)=5), we have (g^{\prime}(5)=\frac{1}{5}).

Answer:

(g(5)=1) (g^{\prime}(5)=\frac{1}{5})