Chapter 4 | Applications of Derivatives
433
and x =2 are possible test points as shown in the following table. Interval Test Point Sign of f ″ ( x ) = 2 ( x −1 ) 3
Conclusion
(−∞, 1)
x =0
+/− =−
f is concave down.
(1, ∞)
x =2
+/+ = +
f is concave up.
From the information gathered, we arrive at the following graph for f .
Find the oblique asymptote for f ( x ) = ⎛
⎝ 3 x 3 −2 x +1 ⎞ ⎠ ⎛ ⎝ 2 x 2 −4 ⎞ ⎠
4.29
.
Example 4.31 Sketching the Graph of a Function with a Cusp
Sketch a graph of f ( x ) = ( x −1) 2/3 .
Solution Step 1. Since the cube-root function is defined for all real numbers x and ( x −1) 2/3 = ⎛ ⎝ x −1 3
⎞ ⎠
2 , the domain
of f is all real numbers. Step 2: To find the y -intercept, evaluate f (0). Since f (0) =1, the y -intercept is (0, 1). To find the x -intercept, solve ( x −1) 2/3 =0. The solution of this equation is x =1, so the x -intercept is (1, 0).
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