Answer
5
Work Step by Step
See Section 3-2 (page 356) on Turning Points of Polynomial Functions.
(If f is degree n, the graph has at most n-1 turning points)
Also see The Leading Coefficient Test (p. 350, end behavior )
( Odd $(\swarrow,\nearrow)$ or $(\nwarrow,\searrow)\quad$Even: $(\nwarrow,\nearrow)$ or $(\swarrow, \searrow)$ )
Also, see Multiplicity and x-lntercepts
(odd: crosses the x-axis, even: touches and turns)
... graphs tend to flatten out near zeros with multiplicity greater than one.
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This graph turns twice: n is at least 3.
End behavior $(\nwarrow,\searrow)$: n is odd.
Three zeros, all of odd multiplicity (crossing the x-axis).
Note the flattening around x=0 ... multiplicity greater than 1 (at least 3)
Each zero, with its multiplicity, generates linear factors of f.
A polynomial of degree n can not have more than n linear factors,
(The Linear Factorization Theorem),
so, n is at least 1+3+1 =5.
