2. [11/13 Points] DETAILS PREVIOUS ANSWERS LARCALCET7 3.6.005.MI.SA. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Verify that f has an inverse function. Then use the function f and the given real number a to find (F-1)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) f(x) = x3 - 2, a = 123 Step 1 The given function is f(x) = x3 - 2. Differentiate f(x) with respect to x. F ( x ) = 3 0 8 3 * 2 6 2 2 Step 2 The function f'(x) = 3x2 is always non-negative non-negative on the domain of f(x). Therefore, f(x) is is monotonic on the domain of x and the inverse of the function exists. Step 3 To find f-1(x), let y = f(x) and solve for x, y = x3 - 2 * = (y + 2 ) 1/3 1/3 Interchange x and y in the expression for x. y = (x+ 2) 1/3 1/3 Find f-1(x) by replacing y with f-1(x). f-1 ( x ) = (x+ 2) 21 1/3 1/3Step 4 To find (f-1)'(x) use the theorem that states that if fis differentiable on an interval I and fhas an inverse function g, that is g = f, then g is differentiable at any x for which f'(g(x)) = 0 and g'(x) = 7 f'(g ( x ) )' Find the values of f-1(123) and then f'(f-1(123)), F- 1 ( x ) = ( x+ 2) 1/3 f-1(123) = 5 $ 5 F' ( F - 1 (123 ) ) = 1 50 9 5 = 3 5 2 = 75 75 Step Substitute the above values in the expression for (f )'(123) = F.(F-1(123)) (F-1)'(123) = f'(f-1(123)) Thus, (f-1)'(123) = Submit Skip (you cannot come back) Need Help? Read It