Question

answer the following true or false:
let $f(x)$ be a differentiable function, such that $f(-3) = 5$ and $f(0) = 5$. then there exists a $c$ with $-3 \leq c \leq 0$ such that the tangent line to $y = f(x)$ is horizontal at $x = c$.
\bigcirc true
\bigcirc false

Explanation:

Step1: Recall Rolle's Theorem

Rolle's Theorem states that if a function \( f(x) \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a)=f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f^{\prime}(c) = 0 \). A horizontal tangent line has a slope of 0, so we need to check if Rolle's Theorem applies here.

Step2: Check Conditions for Rolle's Theorem

• The function \( f(x) \) is given as differentiable. Since differentiability implies continuity, \( f(x) \) is continuous on \([-3, 0]\) (because differentiable functions are continuous on their domain).
• We are given that \( f(-3)=5 \) and \( f(0) = 5 \), so \( f(-3)=f(0) \).
• The interval is \([-3, 0]\), so \( a=-3 \) and \( b = 0 \).

By Rolle's Theorem, there exists a \( c\in(-3, 0)\subseteq[-3, 0] \) such that \( f^{\prime}(c)=0 \). A derivative of 0 at \( x = c \) means the tangent line to \( y = f(x) \) at \( x = c \) is horizontal.

Answer:

True