Find the critical numbers of the function f(x) = x⁸(x - 3)⁷

Solution:

To find the critical numbers of the function it needs to be differentiated then equated to zero. Given f(x) = x⁸(x - 3)⁷

Differentiating the above function we have,

\(\frac{\mathrm{d} x^{8}(x-3)^{7}}{\mathrm{d} x}= x^{8}\frac{\mathrm{d} (x-3)^{7}}{\mathrm{d} x}+(x-3)^{7}\frac{\mathrm{d} x^{8}}{\mathrm{d} x}\)

\(\frac{\mathrm{d} x^{8}(x-3)^{7}}{\mathrm{d} x}= x^{8}(7)(x-3)^{6}(1)+(x-3)^{7}(8)(x^{7})\)

\(\frac{\mathrm{d} x^{8}(x-3)^{7}}{\mathrm{d} x}= 7x^{8}(x-3)^{6}+8x^{7}(x-3)^{7}\)

To determine the Critical points \(\frac{\mathrm{d} x^{8}(x-3)^{7}}{\mathrm{d} x}\) is equated to zero as shown below:

\(\frac{\mathrm{d} x^{8}(x-3)^{7}}{\mathrm{d} x} = 0\)

Therefore we have:

\( 7x^{8}(x-3)^{6}+8x^{7}(x-3)^{7}\) = 0

\(x^{7}(x-3)^{6}[7x + 8(x-3)] = 0\)

\(x^{7}(x-3)^{6}[15x - 24] = 0\) Which implies:

x⁷ = 0 ⇒ x = 0

(x - 3)⁶ = 0 ⇒ x = 3

(15x - 24) = 0 ⇒ x = (24/15) = 8/5

Therefore the critical numbers are x = 0, 3, and 8/5

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Find the critical numbers of the function f(x) = x⁸(x - 3)⁷

Summary:

The critical numbers of the function f(x) = x⁸(x - 3)⁷ are x = 0, 3, and 8/5