(Solved): USE MAPLE ONLY   Cusp and Corner The graph of \( \mathrm{f}(\mathrm{x}) \) has a cusp at \( ...

USE MAPLE ONLY

 

Cusp and Corner The graph of \( \mathrm{f}(\mathrm{x}) \) has a cusp at \( (a, f(a)) \) if \( \mathrm{f}(\mathrm{x}) \) is continuous at 'a' and if the following two conditions hold: (i) \( \quad f^{\prime}(x) \rightarrow \infty \) as \( x \) approaches a from one side (ii) \( \quad f^{\prime}(x) \rightarrow-\infty \) as \( x \) approaches a from the other side The graph of \( \mathrm{f}(\mathrm{x}) \) has a corner at \( (a, f(a)) \) if \( \mathrm{f}(\mathrm{x}) \) is continuous at 'a' and if the right-hand and left-hand derivatives at ' \( \mathrm{a} \) ' exist and are unequal. Consider \( f(x)=|x|+1 \) and \( g(x)=1+x^{\frac{2}{3}} \). Determine the nature of the graphs (is there a cusp, comer, or neither) of \( \mathrm{f}(\mathrm{x}) \) and \( \mathrm{g}(\mathrm{x}) \) near the point \( (0,1) \). HINT: Plot both functions using a proper viewing window as well as the proper graphs. Both graphs is symmetric about the \( \mathrm{x} \)-axis (this is important). Find the derivatives of both functions and then the limits from the left and right at \( x=0 \) of these two derivatives functions. Be sure to state whether each function will have a cusp or a corner or both.