$$ 27 f(x)=x^{3}-x^{2}+4 $$ $$ \begin{array}{ll} ext { (a) Find } f^{\prime}(x) & ext { (b) Evaluate } f^{\prime}(-5)\end{array} $$

Question

$$ 27 f(x)=x^{3}-x^{2}+4 $$ $$ \begin{array}{ll}	ext { (a) Find } f^{\prime}(x) & 	ext { (b) Evaluate } f^{\prime}(-5)\end{array} $$

UpStudy AI · Accepted Answer

Let's solve the given problem step by step.

Given:
$$
27 f(x) = x^3 - x^2 + 4
$$
To find $$ f(x) $$, divide both sides by 27:
$$
f(x) = \frac{x^3 - x^2 + 4}{27}
$$

### (a) Find $$ f^{\prime}(x) $$

To find the derivative $$ f^{\prime}(x) $$, differentiate $$ f(x) $$ with respect to $$ x $$:

$$
f^{\prime}(x) = \frac{d}{dx} \left( \frac{x^3 - x^2 + 4}{27} \right) = \frac{1}{27} \cdot \frac{d}{dx}(x^3 - x^2 + 4)
$$

Differentiate term by term:

$$
f^{\prime}(x) = \frac{1}{27} \left( 3x^2 - 2x + 0 \right) = \frac{3x^2 - 2x}{27}
$$

This can be simplified by factoring the numerator:

$$
f^{\prime}(x) = \frac{x(3x - 2)}{27}
$$

### (b) Evaluate $$ f^{\prime}(-5) $$

Now, substitute $$ x = -5 $$ into the derivative:

$$
f^{\prime}(-5) = \frac{3(-5)^2 - 2(-5)}{27}
$$

Calculate the values:

$$
f^{\prime}(-5) = \frac{3(25) + 10}{27} = \frac{75 + 10}{27} = \frac{85}{27}
$$

So, the final answers are:

**(a)** $$ f^{\prime}(x) = \frac{3x^2 - 2x}{27} $$

**(b)** $$ f^{\prime}(-5) = \frac{85}{27} $$
**(a)** $$ f^{\prime}(x) = \frac{3x^2 - 2x}{27} $$ **(b)** $$ f^{\prime}(-5) = \frac{85}{27} $$

\( 27 f(x)=x^{3}-x^{2}+4 \) \( \begin{array}{ll}\text { (a) Find } f^{\prime}(x) & \text { (b) Evaluate } f^{\prime}(-5)\end{array} \)