Vertical tangent lines? If a function f is continuous at a and limx?a (x)| = ? , then the curve y= f(x) has a vertical tangent line at a and the equation of the tangent line is x = a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 59-60) is used. Use this definition to answer the following questions. Graph the following functions and determine the location of the vertical tangent lines. a.? f(x) = (x ? 2)1/3 b.? f(x) = (x + 1)2/3 c.? f(x) = ?|x?4| d.? f(x) = x5/3 ? 2x1/3

SOLUTION If a function f is continuous at a and lim|xa (x)| = , then the curve y= f(x) has a vertical tangent line at a and the equation of the tangent line is x = a A function is said to be continuous if it satisfies the condition limf(x) = f(a) xa STEP 1 3 (a).Given function f(x) = (x + 1) By definition of continuity the given function is continuous. 2 21 2 1 f (x) = (3 + 1) 3 = (3 + 1) 3 Then, 2 3 x1|f (x)| = x1(x3+ 1) = Thus,the graph of f(x) will be having a vertical tangent line at x=-1. The graph is STEP 2 3 (b).Given function f(x) = (x 2) By definition of continuity the given function is continuous. 1 2 f (x) = (x 2) 31 =...