calculus

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Find an interval of length
0.1 in [1, 2] containing a root of the equation. (Enter your answer using interval notation.)

x7 + 7x − 10 = 0

  • calculus -

    f(1) = -2
    f(2) = 134

    so there is indeed at least one root in [1,2]

    f'(x) = 7x^6 + 7
    f'(x) > 0 everywhere, so f(x) is strictly increasing.

    Since f(2) is so much greater than f(1), I'd expect to find the root near 1

    f(1) = -2
    f(1.05) = -1.243
    f(1.10) = -0.352
    f(1.15) = +0.710

    so, there is a root in [1.05,1.15]

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