Find an interval of length

Question

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

x7 + 7x − 10 = 0

Answer 1

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]