If the area bounded by the curve $y=x-x^{2}$ and line $y=mx$ is $9/2$, then m is equal to:

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to analyze the problem involving the area bounded by the curve $ y = x - x^2 $ and the line $ y = mx $.

### Explanation:

1. **Intersection Points:**
   - The intersection points of the curve $ y = x - x^2 $ and the line $ y = mx $ are found by setting the equations equal: 
     $$
     x - x^2 = mx \implies x - x^2 - mx = 0 \implies x(1 - x - m) = 0
     $$
   - This gives the solutions $ x = 0 $ and $ x = 1 - m $.

2. **Area Calculation:**
   - The area between the curve and the line from $ x = 0 $ to $ x = 1 - m $ is given by the integral:
     $$
     \text{Area} = \int_{0}^{1-m} [(x - x^2) - mx] \, dx = \int_{0}^{1-m} (x - x^2 - mx) \, dx
     $$
   - Simplifying the integrand:
     $$
     x - x^2 - mx = x(1 - x - m)
     $$
   - The integral becomes:
     $$
     \int_{0}^{1-m} (x - x^2 - mx) \, dx = \int_{0}^{1-m} (x - x^2 - mx) \, dx = \int_{0}^{1-m} (x - x^2 - mx) \, dx
     $$
   - Evaluating the integral:
     $$
     \int_{0}^{1-m} (x - x^2 - mx) \, dx = \left[ \frac{x^2}{2} - \frac{x^3}{3} - \frac{mx^2}{2} \right]_{0}^{1-m}
     $$
   - Substituting the limits:
     $$
     \left( \frac{(1-m)^2}{2} - \frac{(1-m)^3}{3} - \frac{m(1-m)^2}{2} \right) - \left( 0 \right)
     $$
   - Simplifying further, the area is set equal to $ \frac{9}{2} $.

3. **Solving for $ m $:**
   - Solving the equation derived from the area condition will yield the values of $ m $ that satisfy the area being $ \frac{9}{2} $.
   - Through solving, it is found that $ m = -2 $ and $ m = 4 $ satisfy the condition.

### Justification for Option 4:

- **Option 4: -2, 4** is correct because both values of $ m $ satisfy the condition that the area between the curve and the line is $ \frac{9}{2} $.

### Explanation for Incorrect Options:

- **Option 1: -4 only** is incorrect because $ m = -4 $ does not satisfy the area condition.
- **Option 2: -2 only** is incorrect because while $ m = -2 $ is a correct solution, it is not the only solution.
- **Option 3: 2, -4** is incorrect because neither $ m = 2 $ nor $ m = -4 $ satisfies the area condition.

Thus, Option 4 is the correct choice as it includes both values $ m = -2 $ and $ m = 4 $, which satisfy the given area condition.