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Therefore, by the second derivative test x 5 gives a local maximum and x 1
9
gives a local minimum. Since this is a polynomial function, f 1x2 must be decreasing
when x 6 1, increasing when 1 6 x 6 5 and decreasing when x 7 5 . For a
9
9
point of inflection, f –1x2 0 and changes sign.
18x 4 0 or x 2 9
Now we determine the sign of f –1x2 in, the intervals determined by x 2 9 .
A point of inflection occurs at about 10.2, 1.22.
We can now draw our sketch.
Interval x 6 2 9
x 2 9
x 7 2 9
f (x) 7 0 0 6 0
Graph of f(x) concave up point of inflection concave down
6
4
2
–6 –4 –2 0
–2
minimum (–1, –4)
–4
–6
y
maximum (0.6, 1.6)
x
2 4 6
point of
inflection
(–0.2, –1.2)
EXAMPLE 2
Sketching an accurate graph of a rational function
Sketch the graph of f 1x2 x 4
x 2 x 2 .
Solution
Analyze f (x).
f 1x2 is a rational function.
Determine any intercepts.
x-intercept, y 0 y-intercept, x 0
x 4
y 0 4
x 2 x 2 0 0 0 2
x 4 0 y 4
2
x 4 y 2
14, 02 10, 22
NEL
CHAPTER 4 209