Question 7.3.2: Suppose the function f is defined for all real x by the foll......
Suppose the function f is defined for all real x by the following formula: f (x) = x^{5} + 3x^{3} + 6x − 3. Show that f has an inverse function g, and then, given that f (1) = 7, use formula (7.3.2) to find g^{\prime}(7).
g^{\prime}(y_{0})={\frac{1}{f^{\prime}(x_{0})}} (7.3.2)
Differentiating f (x) yields f^{\prime}(x) = 5x^{4} + 9x^{2} + 6. Clearly, f^{\prime}(x) > 0 for all x, so f is strictly increasing and consequently it is one-to-one. It therefore has an inverse function g. To find g^{\prime}(7), we use formula (7.3.2) with x_{0} = 1 and y_{0} = 7. Since f^{\prime}(1) = 20, we obtain g^{\prime}(7) = 1/ f^{\prime} (1) = 1/20. Note that we have found g^{\prime}(7) exactly even though it is impossible to find any algebraic formula for the inverse function g.