Calculus (3rd Edition)

Published by W. H. Freeman

Chapter 3 - Differentiation - 3.2 The Derivative as a Function - Exercises - Page 115: 52

Answer

There are parallel tangent lines at $x=-1, x=\frac{5}{3}$.

Work Step by Step

To have parallel tangent lines, we must have equal slopes; that is, $$\frac{dy}{dx}=3x^2, \quad \frac{dy}{dx}=2x+5,$$ then $3x^2=2x+5$. We get the equation $$3x^2-2x-5=0\Longrightarrow (3x-5)(x+1)=0.$$ There are parallel tangent lines at $x=-1, x=\frac{5}{3}$.

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