determine if the following piecewise - defined function is differentiable at x = 0.\n f(x)=\begin{cases}2x…

determine if the following piecewise - defined function is differentiable at x = 0.\n f(x)=\begin{cases}2x - 3, & xgeq0\\x^{2}+3x - 3, & x < 0end{cases}\nselect the correct choice below and, if necessary, fill in the answer boxes within your choice\na. the function is differentiable at x = 0 because it is continuous at x = 0 and (lim_{h\rightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=) and (lim_{h\rightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=)\n(type integers or simplified fractions.)\nb. the function is not differentiable at x = 0 because it is continuous at x = 0 and (lim_{h\rightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=) and (lim_{h\rightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=)\n(type integers or simplified fractions.)\nc. the function is not differentiable at x = 0 because it is not continuous at x = 0.

determine if the following piecewise - defined function is differentiable at x = 0.\n f(x)=\begin{cases}2x - 3, & xgeq0\\x^{2}+3x - 3, & x < 0end{cases}\nselect the correct choice below and, if necessary, fill in the answer boxes within your choice\na. the function is differentiable at x = 0 because it is continuous at x = 0 and (lim_{h\rightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=) and (lim_{h\rightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=)\n(type integers or simplified fractions.)\nb. the function is not differentiable at x = 0 because it is continuous at x = 0 and (lim_{h\rightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=) and (lim_{h\rightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=)\n(type integers or simplified fractions.)\nc. the function is not differentiable at x = 0 because it is not continuous at x = 0.

Answer

Explanation:

Step1: Find the left - hand limit of the difference quotient

For (x<0), (f(x)=x^{2}+3x - 3). First, find (f(0)) using the part of the function for (x\geq0), so (f(0)=2\times0 - 3=-3). The left - hand limit of the difference quotient (\lim_{h\rightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=\lim_{h\rightarrow0^{-}}\frac{(h^{2}+3h - 3)-(-3)}{h}=\lim_{h\rightarrow0^{-}}\frac{h^{2}+3h}{h}=\lim_{h\rightarrow0^{-}}(h + 3)=3).

Step2: Find the right - hand limit of the difference quotient

For (x\geq0), (f(x)=2x - 3). The right - hand limit of the difference quotient (\lim_{h\rightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=\lim_{h\rightarrow0^{+}}\frac{(2h-3)-(-3)}{h}=\lim_{h\rightarrow0^{+}}\frac{2h}{h}=2). Since (\lim_{h\rightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=3) and (\lim_{h\rightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=2), the left - hand and right - hand limits of the difference quotient are not equal.

Answer:

B. The function is not differentiable at (x = 0) because it is continuous at (x = 0) and (\lim_{h\rightarrow0^{-}}\frac{f(0 + h)-f(0)}{h}=3) and (\lim_{h\rightarrow0^{+}}\frac{f(0 + h)-f(0)}{h}=2)