For what values of `x`  is the rate of increase of `x^3-5x^2+5x+8`  is twice the rate of increase of `x` ?

Options

A. (a) `-3, -1/3`
B. (b) `-3, 1/3`
C. (c) `3, -1/3`
D. (d) `3, 1/3`

Explanation

To find the values of \( x \) for which the rate of increase of the function \( f(x) = x^3 - 5x^2 + 5x + 8 \) is twice the rate of increase of \( x \), we will follow these steps:

### Step 1: Find the derivative of the function \( f(x) \)

The rate of increase of \( f(x) \) is given by its derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}(x^3 - 5x^2 + 5x + 8) \]

Using the power rule for differentiation:

\[ f'(x) = 3x^2 - 10x + 5 \]

### Step 2: Find the derivative of \( x \)

The rate of increase of \( x \) is simply the derivative of \( x \):

\[ \frac{d}{dx}(x) = 1 \]

### Step 3: Set up the equation

We need to find when the rate of increase of \( f(x) \) is twice the rate of increase of \( x \):

\[ f'(x) = 2 \cdot 1 \]

This simplifies to:

\[ 3x^2 - 10x + 5 = 2 \]

### Step 4: Rearrange the equation

Now, rearranging the equation gives:

\[ 3x^2 - 10x + 5 - 2 = 0 \]

Which simplifies to:

\[ 3x^2 - 10x + 3 = 0 \]

### Step 5: Solve the quadratic equation

We can use the quadratic formula to solve for \( x \):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 3 \), \( b = -10 \), and \( c = 3 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-10)^2 - 4 \cdot 3 \cdot 3 = 100 - 36 = 64 \]

Now substituting into the quadratic formula:

\[ x = \frac{10 \pm \sqrt{64}}{2 \cdot 3} \]
\[ x = \frac{10 \pm 8}{6} \]

This gives us two solutions:

1. \( x = \frac{18}{6} = 3 \)
2. \( x = \frac{2}{6} = \frac{1}{3} \)

### Final Solutions

The values of \( x \) for which the rate of increase of \( f(x) \) is twice the rate of increase of \( x \) are:

\[ x = 3 \quad \text{and} \quad x = \frac{1}{3} \]