Question

closed interval -1,4, where g(-1)=-4 and g(4)=1. which of the following is guaranteed by the intermediate value theorem? choose 1 answer: a g(c)=-3 for at least one c between -4 and 1 b g(c)=3 for at least one c between -1 and 4 c g(c)=3 for at least one c between -4 and 1 d g(c)=-3 for at least one c

Explanation:

Step1: Recall Intermediate - Value Theorem

If a function $g(x)$ is continuous on the closed interval $[a,b]$, and $k$ is a number between $g(a)$ and $g(b)$, then there exists at least one number $c$ in the interval $(a,b)$ such that $g(c)=k$. Here, $a = - 1$, $b = 4$, $g(-1)=-4$ and $g(4)=1$.

Step2: Analyze each option

• Option A: The interval for $c$ should be between $-1$ and $4$ (the domain interval), not between $-4$ and $1$ (the range values), so it is incorrect.
• Option B: Since $3$ is not between $-4$ and $1$, by the Intermediate - Value Theorem, we cannot guarantee that there is a $c$ in $[-1,4]$ such that $g(c)=3$.
• Option C: Similar to Option B, $3$ is not between $-4$ and $1$, so this option is incorrect.
• Option D: Since $-3$ is between $-4$ and $1$, and the function $g(x)$ is continuous on $[-1,4]$, by the Intermediate - Value Theorem, there exists at least one $c$ in the interval $(-1,4)$ such that $g(c)=-3$.

Answer:

D. $g(c)=-3$ for at least one $c$ between $-1$ and $4$