Question

is f(x) = \frac{x^{2}-5x + 6}{x - 3} differentiable at x = 3? justify your answer. choose the correct answer below. a. yes, the function is differentiable because f(x) is not continuous at x = 3. b. no, the function is not differentiable because f(x)=0 at x = 3. c. no, the function is not differentiable because f(x) is not continuous at x = 3. d. yes, the function is differentiable because f(x) is continuous at x = 3.

Explanation:

Step1: Simplify the function

First, factor the numerator $x^{2}-5x + 6=(x - 2)(x - 3)$. So $f(x)=\frac{(x - 2)(x - 3)}{x - 3}=x - 2$ for $x
eq3$. But at $x = 3$, the original function $f(x)=\frac{x^{2}-5x + 6}{x - 3}$ is undefined since the denominator is 0.

Step2: Check continuity

A function $y = f(x)$ is continuous at $x=a$ if $\lim_{x
ightarrow a}f(x)=f(a)$. Here, $\lim_{x
ightarrow3}f(x)=\lim_{x
ightarrow3}(x - 2)=1$, but $f(3)$ is undefined. So $f(x)$ is not continuous at $x = 3$.

Step3: Recall differentiability - continuity relationship

A function must be continuous at a point to be differentiable at that point. Since $f(x)$ is not continuous at $x = 3$, it is not differentiable at $x = 3$.

Answer:

C. No, the function is not differentiable because f(x) is not continuous at x = 3.