Use the Intermediate Value Theorem to determine if a real zero of f(x)occurs in the given interval. If this cannot be
determined, indicate so.
-$f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } + 2 x + 5 \text { in } ( 5,6 )$
A)Yes
B)Cannot determine

Question

Quizplus · Accepted Answer

The Intermediate Value Theorem states that if $f(x)$ is continuous on a closed interval $[a, b]$, and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in the interval $(a, b)$ such that $f(c) = N$. To apply this theorem to $f(x) = -x^3 - 6x^2 + 2x + 5$ in the interval $(5, 6)$, we need to evaluate $f(5)$ and $f(6)$:$f(5) = -(5)^3 - 6(5)^2 + 2(5) + 5 = -125 - 150 + 10 + 5 = -260$$f(6) = -(6)^3 - 6(6)^2 + 2(6) + 5 = -216 - 216 + 12 + 5 = -415$Since $f(5)$ and $f(6)$ have different signs (one is negative and the other is also negative, but we are looking for a change in sign for a real zero to be guaranteed), my explanation was incorrect in suggesting a sign change based solely on the values given. Without a sign change between $f(5)$ and $f(6)$, we cannot directly apply the Intermediate Value Theorem to guarantee a zero in the interval $(5, 6)$. Therefore, the correct interpretation should be that we cannot determine if a real zero occurs in the given interval based solely on this information. My initial conclusion was incorrect, and the correct choice should have been based on a proper analysis of the sign change (or lack thereof) between $f(5)$ and $f(6)$.

Use the Intermediate Value Theorem to Determine If a Real f(x)=−x3−6x2+2x+5 in (5,6)f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } + 2 x + 5 \text { in } ( 5,6 )f(x)=−x3−6x2+2x+5 in (5,6)

Use the Intermediate Value Theorem to Determine If a Real $f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } + 2 x + 5 \text { in } ( 5,6 )$