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Chapter 2

Q. 2.3

Find all horizontal and vertical tangents to the curve of Example 2.

Step-by-Step

Verified Solution

There are vertical tangents when

f_{1}^{\prime}(t)=6 t^{2}-30 t+24=0=6\left(t^{2}-5 t+4\right)=6(t-4)(t-1),

or when t = 1 and t = 4, since f_{2}^{\prime}(t) is nonzero at these points. When t = 1,  x = 18; and when t = 4, x = -9. Thus the vertical tangents are the lines x = 18 and x = -9. There is a horizontal tangent when 2 t + 1 = 0, or t=-\frac{1}{2}. When t=-\frac{1}{2}, y=\frac{3}{4}; and the line y=\frac{3}{4} is a horizontal tangent.