Question

use the intermediate value theorem to show that f has a zero between a and b.

$f(x)=3x^{4}+3x - 3$; $a = 0$, $b = 1$

$f(a)=

$f(b)=

since $f(a) ? 0$ and $f(b) ? 0$, we see that $f(c)=0$ for at least one real number c between a and b.

Explanation:

Step1: Calculate f(a)

Substitute \(a = 0\) into \(f(x)=3x^{4}+3x - 3\).
\[f(0)=3\times0^{4}+3\times0 - 3=- 3\]

Step2: Calculate f(b)

Substitute \(b = 1\) into \(f(x)=3x^{4}+3x - 3\).
\[f(1)=3\times1^{4}+3\times1 - 3=3 + 3-3=3\]

Answer:

\(f(a)=-3\), \(f(b)=3\), since \(f(a)<0\) and \(f(b)>0\)