Use the intermediate value theorem to show that f(x) = 2x³ − 3 has a zero in the interval [1, 2].

The function $f(x) = 2x^3 - 3$ is a polynomial function. Polynomials are continuous for all real numbers, and therefore, $f(x)$ is continuous on the closed interval $[1, 2]$.

Calculate $f(1)$:

Calculate $f(2)$:

We have $f(1) = -1$ and $f(2) = 13$. Since $f(1)$ is negative and $f(2)$ is positive, and $f(x)$ is continuous on $[1, 2]$, the Intermediate Value Theorem states that there must exist at least one value $c$ in the open interval $(1, 2)$ such that $f(c) = 0$.
Since $f(c) = 0$ means that $c$ is a zero of the function, we have shown that $f(x) = 2x^3 - 3$ has a zero in the interval $[1, 2]$.