Question

assignment 4: problem 14 (1 point) let (f(x)=\begin{cases}x^{2}-4x&xleq6\\2x&x > 6end{cases}) (a) is (f) continuous at (x = 6)? (b) is (f) differentiable at (x = 6)? (c) if (f) is differentiable at (x = 6) enter the value of (f(6)). if (f) is not differentiable at (x = 6) enter the value of (f(6)) instead. (f(6)) or (f(6)=)

Explanation:

Step1: Check left - hand limit

For $x\leq6$, $f(x)=x^{2}-4x$. Calculate $\lim_{x
ightarrow6^{-}}f(x)=\lim_{x
ightarrow6^{-}}(x^{2}-4x)=6^{2}-4\times6=36 - 24=12$.

Step2: Check right - hand limit

For $x > 6$, $f(x)=2x$. Calculate $\lim_{x
ightarrow6^{+}}f(x)=\lim_{x
ightarrow6^{+}}2x=2\times6 = 12$.

Step3: Calculate function value

$f(6)=6^{2}-4\times6=12$. Since $\lim_{x
ightarrow6^{-}}f(x)=\lim_{x
ightarrow6^{+}}f(x)=f(6)=12$, $f(x)$ is continuous at $x = 6$.

Step4: Check left - hand derivative

For $x\leq6$, $f(x)=x^{2}-4x$, $f'(x)=2x - 4$. Then $f'(6^{-})=2\times6-4 = 8$.

Step5: Check right - hand derivative

For $x>6$, $f(x)=2x$, $f'(x)=2$. Then $f'(6^{+})=2$. Since $f'(6^{-})
eq f'(6^{+})$, $f(x)$ is not differentiable at $x = 6$.

Step6: Find the required value

Since $f(x)$ is not differentiable at $x = 6$, we enter $f(6)$. And $f(6)=12$.

Answer:

12