AP Calculus Derivatives of Inverse Functions?

Verify that f has an inverse. Then use the function f and the given real number a to find

(f ^−1)'(a).

f(x) = 1/27(x^5 + 2x^3) a = −11

Find (f^-1)'(-11)

1 Answer

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  • kb
    Lv 7
    3 years ago

    Observe that f'(x) = (1/27)(5x^4 + 6x^2) is non-decreasing. Hence, f has an inverse.

    When a (= y) = -11, we find that x = -3, because f(-3) = -11.

    So, the Inverse Function Theorem yields

    (f^(-1))'(-11) = 1/f'(x) {at x = -3}

    ....................= 1/[(1/27)(5x^4 + 6x^2)] {at x = -3}

    ....................= 27/(x^2 * (5x^2 + 6)) {at x = -3}

    ....................= 27/(9 * (45 + 6))

    ....................= 1/17.

    I hope this helps!

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