attempt 1: 2 attempts remaining. let f(x)=5x^1/3(x^3 - 10). evaluate the following specific values of f: (a)…

Explanation:

Step1: Expand the function

First, expand $f(x)=5x^{1/3}(x^{3}-10)=5x^{1/3 + 3}-50x^{1/3}=5x^{10/3}-50x^{1/3}$.

Step2: Differentiate using power - rule

The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$. So, $f'(x)=5\times\frac{10}{3}x^{\frac{10}{3}-1}-50\times\frac{1}{3}x^{\frac{1}{3}-1}=\frac{50}{3}x^{7/3}-\frac{50}{3}x^{-2/3}$.

Step3: Evaluate $f'(3)$

Substitute $x = 3$ into $f'(x)$: [ \begin{align*} f'(3)&=\frac{50}{3}\times3^{7/3}-\frac{50}{3}\times3^{-2/3}\ &=\frac{50}{3}\times3^{7/3}-\frac{50}{3\times3^{2/3}}\ &=\frac{50\times3^{7/3}\times3^{2/3}-50}{3\times3^{2/3}}\ &=\frac{50\times3^{3}-50}{3\times3^{2/3}}\ &=\frac{50\times27 - 50}{3\times3^{2/3}}\ &=\frac{50(27 - 1)}{3\times3^{2/3}}\ &=\frac{1300}{3\times3^{2/3}} \end{align*} ]

Step4: Evaluate $f'(5)$

Substitute $x = 5$ into $f'(x)$: [ \begin{align*} f'(5)&=\frac{50}{3}\times5^{7/3}-\frac{50}{3}\times5^{-2/3}\ &=\frac{50\times5^{7/3}- \frac{50}{5^{2/3}}}{3}\ &=\frac{\frac{50\times5^{3}-50}{5^{2/3}}}{3}\ &=\frac{50(125 - 1)}{3\times5^{2/3}}\ &=\frac{6200}{3\times5^{2/3}} \end{align*} ]

Answer:

(A) $\frac{1300}{3\times3^{2/3}}$ (B) $\frac{6200}{3\times5^{2/3}}$