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.
-$f ( x ) = x ^ { 5 } + 2 x ^ { 4 } - 2 x ^ { 3 } + 2 x ^ { 2 } - 5 x - 3 \text { in } ( - 1,0 )$
A)Cannot determine
B)Yes

Question

Quizplus · Accepted Answer

The Intermediate Value Theorem states that if $f$ 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, we evaluate $f(-1)$ and $f(0)$:- $f(-1) = (-1)^5 + 2(-1)^4 - 2(-1)^3 + 2(-1)^2 - 5(-1) - 3 = -1 + 2 + 2 + 2 + 5 - 3 = 7$- $f(0) = 0^5 + 2(0)^4 - 2(0)^3 + 2(0)^2 - 5(0) - 3 = -3$Since $f(-1) = 7$ and $f(0) = -3$, and these values have opposite signs, by the Intermediate Value Theorem, there must be at least one real zero of $f(x)$ in the interval $(-1, 0)$.

Use the Intermediate Value Theorem to Determine If a Real f(x)=x5+2x4−2x3+2x2−5x−3 in (−1,0)f ( x ) = x ^ { 5 } + 2 x ^ { 4 } - 2 x ^ { 3 } + 2 x ^ { 2 } - 5 x - 3 \text { in } ( - 1,0 )f(x)=x5+2x4−2x3+2x2−5x−3 in (−1,0)

Use the Intermediate Value Theorem to Determine If a Real $f ( x ) = x ^ { 5 } + 2 x ^ { 4 } - 2 x ^ { 3 } + 2 x ^ { 2 } - 5 x - 3 \text { in } ( - 1,0 )$