Question

does f(x)=2 + x - x^2 holds with ivt given 0,3; w = 1? ivt holds no sufficient information ivt does not hold

Explanation:

Step1: Recall the Intermediate - Value Theorem (IVT)

The IVT states that if a function $y = f(x)$ is continuous on a closed interval $[a,b]$, and $k$ is a number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in the interval $(a,b)$ such that $f(c)=k$. First, check the continuity of $f(x)=2 + x - x^{2}$. Since $f(x)$ is a polynomial, it is continuous on the interval $[0,3]$.

Step2: Calculate $f(0)$ and $f(3)$

$f(0)=2 + 0-0^{2}=2$.
$f(3)=2 + 3-3^{2}=2 + 3 - 9=-4$.

Step3: Check if $w = 1$ is between $f(0)$ and $f(3)$

Since $-4<1<2$, by the Intermediate - Value Theorem, there exists a $c\in(0,3)$ such that $f(c) = 1$.

Answer:

IVT holds