Question

how many real solutions does the equation y=−x^3−x^2+5x+2 have?

Answer 1

To find the number of real solutions to the equation \( y = -x^3 - x^2 + 5x + 2 \), we can analyze the function \( f(x) = -x^3 - x^2 + 5x + 2 \) and determine its behavior.

1. **Finding the derivative**:
We first calculate the derivative to understand the function's behavior:
\[
f'(x) = -3x^2 - 2x + 5
\]

2. **Finding critical points**:
Next, we set the derivative equal to zero to find the critical points:
\[
-3x^2 - 2x + 5 = 0
\]
To solve this quadratic equation, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = -3 \), \( b = -2 \), and \( c = 5 \):
\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(5)}}{2(-3)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 60}}{-6}
\]
\[
x = \frac{2 \pm \sqrt{64}}{-6}
\]
\[
x = \frac{2 \pm 8}{-6}
\]
This gives us two critical points:
\[
x = \frac{10}{-6} = -\frac{5}{3} \quad \text{and} \quad x = \frac{-6}{-6} = 1
\]

3. **Evaluating the second derivative**:
To determine the nature of these critical points, we can evaluate the second derivative:
\[
f''(x) = -6x - 2
\]
Substituting the critical points into \( f''(x) \):
- At \( x = -\frac{5}{3} \):
\[
f''\left(-\frac{5}{3}\right) = -6\left(-\frac{5}{3}\right) - 2 = 10 - 2 = 8 > 0 \quad \text{(local minimum)}
\]
- At \( x = 1 \):
\[
f''(1) = -6(1) - 2 = -6 - 2 = -8 < 0 \quad \text{(local maximum)}
\]

4. **Finding values at critical points**:
Now, we evaluate \( f(x) \) at these critical points:
- At \( x = -\frac{5}{3} \):
\[
f\left(-\frac{5}{3}\right) = -\left(-\frac{5}{3}\right)^3 - \left(-\frac{5}{3}\right)^2 + 5\left(-\frac{5}{3}\right) + 2
\]
Calculating this:
\[
= -\left(-\frac{125}{27}\right) - \frac{25}{9} - \frac{25}{3} + 2
\]
(You can compute this to find the approximate value.)

- At \( x = 1 \):
\[
f(1) = -1 - 1 + 5 + 2 = 5
\]

5. **Behavior as \( x \to \pm \infty \)**:
As \( x \to -\infty \), \( f(x) \to +\infty \) because the leading term is negative. As \( x \to +\infty \), \( f(x) \to -\infty \).

By examining the local maximum and minimum, we can infer:
- \( f(x) \) starts high, dips down to a local minimum, rises to a local maximum, and then falls again. This suggests that there are three real roots since the graph crosses the x-axis three times.

Therefore, the equation \( y = -x^3 - x^2 + 5x + 2 \) has **three real solutions**.