To determine how many real solutions the equation \( y = -x^3 - x^2 + 5x + 2 \) has, we generally look for the points where the curve intersects the x-axis (where \( y = 0 \)).
Given that you have a graph of the curve, you would count the number of times the curve crosses the x-axis. Each crossing represents a real solution to the equation.
Without the actual graph, I can help you analyze the function.
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Behavior at Infinity: The function is a cubic polynomial and the leading term is negative, meaning that as \( x \) approaches positive infinity, \( y \) approaches negative infinity, and as \( x \) approaches negative infinity, \( y \) approaches positive infinity.
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Critical Points: You can find critical points by taking the derivative of \( y \) and setting it equal to zero, which allows you to find local maxima and minima. This helps in understanding changes in direction.
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Descarte's Rule of Signs: Analyze the number of sign changes in \( y \) when you substitute different values of \( x \) to determine the possible number of positive and negative roots.
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Values of y: Evaluate \( y \) for several values of \( x \) (for example, -10 to 10) to get a better sense of the behavior of the function throughout the given range.
However, without the actual graph or the ability to visualize it right now, I can't give you the exact number of real solutions. Typically, for cubic functions, you can expect either one or three real solutions, depending on whether the curve has turning points that cross the x-axis.
If you have the graph:
- Count the number of times the curve intersects the x-axis.
- If it intersects more than once, there are possibly three solutions; if it only touches and doesn't cross, it's one solution.
Let me know if you have the graph or any further details, and I'd be glad to help!