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# find the critical numbers of f(x) = X^4 (x – 2)³?

applications of differentiation

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- Lazy Eight (∞)Lv 51 decade agoFavorite Answer
Given f(x) = x^4(x-2)³

Take the derivative of f(x) using product rule:

f '(x) = 4x³ * (x - 2)³ + 3(x-2)² * x^4

Simplify (I will let you do this since this only algebra)

f '(x) = 7x^6 - 36x^5 + 60x^4 - 32x^3

set f ' (x) = 0

7x^6 - 36x^5 + 60x^4 - 32x^3 = 0

Factor out x³

x³(7x³ - 36x² + 60x - 32) = 0

x³ = 0 or 7x³ - 36x² + 60x - 32 = 0

x = 0 <==== That's one of your critical numbers

Now, we have to solve 7x³ - 36x² + 60x - 32 = 0

Solving the above equation gives (I will let you do that, you can also use mathematical software to find those zeros)

x = 2 and x = 8/7

Therefore, the critical numbers are x = 0, 8/7, 2.

Source(s): LZY8 Math Professor

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