Calculus (3rd Edition)

$[1.25, 1.5]$
$f(x)= x^{7}+3x-10$ is continuous as it is a polynomial function. This equation will have a root in an interval with the endpoints having nonzero values with opposite signs. Let us split the given interval [1,2] into four intervals so that each interval will have a length of $\frac{1}{4}$. The intervals obtained are: [1,1.25], [1.25,1.5], [1.5,1.75], [1.75,2] Now, let us check if any of the intervals have endpoints with nonzero values and opposite signs. $f(1)=1+3-10=-6$ $f(1.25)= (1.25)^{7}+3(1.25)-10=-1.481628$ The value of the function at both endpoints will have the same sign; therefore, $[1,1.25]$ does not contain the root. $f(1.5)= (1.5)^{7}+3(1.5)-10= 11.58594$ As the values of the function at the endpoints of the interval [1.25,1.5] are opposite in sign and are nonzero, according to intermediate value theorem, this continuous function should have a root in [1.25,1.5].