practice another\ndetermine whether the function is differentiable at x = 3.\nf(x) = {x^2 + 2, x ≤ 3; 6x…

practice another\ndetermine whether the function is differentiable at x = 3.\nf(x) = {x^2 + 2, x ≤ 3; 6x - 9, x > 3}\nthe function is differentiable at x = 3.\nthe function is not differentiable at x = 3.

practice another\ndetermine whether the function is differentiable at x = 3.\nf(x) = {x^2 + 2, x ≤ 3; 6x - 9, x > 3}\nthe function is differentiable at x = 3.\nthe function is not differentiable at x = 3.

Answer

Explanation:

Step1: Check left - hand derivative

For $x\leq3$, $f(x)=x^{2}+2$. The derivative $f^\prime(x) = 2x$. The left - hand derivative at $x = 3$ is $f^\prime_{-}(3)=2\times3 = 6$.

Step2: Check right - hand derivative

For $x>3$, $f(x)=6x - 9$. The derivative $f^\prime(x)=6$. The right - hand derivative at $x = 3$ is $f^\prime_{+}(3)=6$.

Step3: Compare left and right derivatives

Since $f^\prime_{-}(3)=f^\prime_{+}(3) = 6$, the function is differentiable at $x = 3$.

Answer:

The function is differentiable at $x = 3$.