To determine the number of real solutions of the equation \( f(x) = -2x^2 - 1 \), we can analyze its graph.
The function \( f(x) = -2x^2 - 1 \) is a quadratic function in the standard form \( f(x) = ax^2 + bx + c \), where \( a = -2 \), \( b = 0 \), and \( c = -1 \).
- Since the coefficient of \( x^2 \) (which is \( a = -2 \)) is negative, the parabola opens downwards.
- The vertex of the parabola can be found using the formula \( x = -\frac{b}{2a} \). In this case: \[ x = -\frac{0}{2 \times -2} = 0. \]
- We can find the value of \( f(0) \): \[ f(0) = -2(0)^2 - 1 = -1. \]
Thus, the vertex of the parabola is at \( (0, -1) \).
Since the parabola opens downwards and the vertex at \( (0, -1) \) is above the x-axis, the entire graph lies below the x-axis. This means the function does not intersect the x-axis at any point.
Therefore, the equation \( f(x) = -2x^2 - 1 = 0 \) has no real solutions.
Conclusion: The number of real solutions is 0.
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