Question

PLSSS HELP State the maximum number of turns the graph of each function could make 1. f(x)=x^5-3x+1 2.f(x)=-x^7-7x^5-4x^3

Answer 1

1. max for 5th-degree: 4 turns. This function: 2 turns.
2. max for 7th-degree: 6 turns. This function: 0 turns.

Explanation:

In general, the graph of an n-th degree function can make n-1 turns. However, in specific cases, the number of turns is limited by the number of real zero-crossings of the derivative.

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1. This 5th-degree function can have at most 4 turns. However, the derivative, f'(x) = 5x^4 -3, has only two (2) real zeros. Hence the graph of this function can only have 2 turns.

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2. This 7th-degree function can have at most 6 turns. However, the derivative, f'(x) = -7x^6 -35x^4-12x^2, has an even-multiplicity root at x=0 only. The derivative never crosses 0. Hence the graph makes no turns.