Question

explain why the function has at least one zero in the given interval.
function interval
$f(x)=x^{3}+7x - 5$ $0,1$
at least one zero exists because $f(x)$ is continuous and $f(0)>0$ while $f(1)<0$.
at least one zero exists because $f(x)$ is not continuous.
at least one zero exists because $f(x)$ being a third - degree polynomial must have a real solution.
at least one zero exists because $f(x)$ is continuous and $f(0)<0$ while $f(1)>0$.
at least one zero exists because $f(x)$ is continuous and $f(0)

Explanation:

The function \( f(x) = x^3 + 7x - 5 \) is a polynomial, hence continuous on \([0, 1]\). Calculate \( f(0) = -5 < 0 \) and \( f(1) = 3 > 0 \). By the Intermediate Value Theorem, since continuous with opposite signs at endpoints, there's at least one zero in \([0, 1]\).

Answer:

At least one zero exists because \( f(x) \) is continuous and \( f(0) < 0 \) while \( f(1) > 0 \).