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Determine all critical points for the function:

\[ f(x) = 45x^3 - 3x^5 \]

A. \( x = 3 \)
B. \( x = -3 \) and \( x = 3 \)
C. \( x = -3 \)
D. \( x = 0 \), \( x = -3 \), and \( x = 3 \)

Answer :

Critical points of the function f(x) = 45x³−3x⁵ are 0 , 3 and -3 .

The correct option is D .

Given, that function f(x) = 45x³−3x⁵ .

To calculate critical points differentiate f(x) with respect to x and equate it to zero.

[tex]f'(x) = 0\\\\\frac{d(45x^3 - 3x^5)}{dx} = 0 \\\\45\times 3(x^2) - 15x^4 = 0\\\\15x^2(9 - x^2) = 0\\[/tex]

Solve for x.

x = 0

x = ±3

Thus the correct option is D .

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