Let `h(x) = f (x) -(f (x))^2+ (f (x))^3` for every real number x: Then

Options

A. h is increasing whenever f is increasing
B. h is increasing whenever f is decreasing
C. h is decreasing whenever f is decreasing
D. nothing can be said in general.

Answer

To solve the problem, we need to analyze the function \( h(x) = f(x) - (f(x))^2 + (f(x))^3 \) and determine when it is increasing or decreasing. We will do this by finding the derivative \( h'(x) \) and analyzing its sign.

### Step 1: Differentiate \( h(x) \)

We start by differentiating \( h(x) \):
\[ h'(x) = \frac{d}{dx}[f(x)] - \frac{d}{dx}[(f(x))^2] + \frac{d}{dx}[(f(x))^3] \]

Using the chain rule, we differentiate each term:
- The derivative of \( f(x) \) is \( f'(x) \).
- The derivative of \( (f(x))^2 \) is \( 2f(x)f'(x) \).
- The derivative of \( (f(x))^3 \) is \( 3(f(x))^2f'(x) \).

Putting it all together, we have:
\[ h'(x) = f'(x) - 2f(x)f'(x) + 3(f(x))^2f'(x) \]

### Step 2: Factor out \( f'(x) \)

Next, we can factor \( f'(x) \) out of the expression:
\[ h'(x) = f'(x) \left( 1 - 2f(x) + 3(f(x))^2 \right) \]

### Step 3: Analyze the quadratic expression

Let \( y = f(x) \). Then we can rewrite the expression in the parentheses:
\[ 1 - 2y + 3y^2 \]

This is a quadratic equation in \( y \). To analyze its sign, we can complete the square or find its roots.

### Step 4: Finding the roots

The roots of the quadratic \( 3y^2 - 2y + 1 = 0 \) can be found using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} = \frac{2 \pm \sqrt{4 - 12}}{6} = \frac{2 \pm \sqrt{-8}}{6} \]

Since the discriminant is negative, the quadratic has no real roots and is always positive (as the coefficient of \( y^2 \) is positive).

### Step 5: Conclusion about \( h'(x) \)

Since \( 1 - 2f(x) + 3(f(x))^2 > 0 \) for all real \( f(x) \), the sign of \( h'(x) \) depends solely on \( f'(x) \):
- If \( f'(x) > 0 \), then \( h'(x) > 0 \) (i.e., \( h(x) \) is increasing).
- If \( f'(x) < 0 \), then \( h'(x) < 0 \) (i.e., \( h(x) \) is decreasing).

### Step 6: Final Result

Thus, we conclude:
- \( h(x) \) is increasing when \( f(x) \) is increasing.
- \( h(x) \) is decreasing when \( f(x) \) is decreasing.