Question

what is the solution set of the quadratic inequality ( f(x) geq 0 )?
( {x | x in mathbb{r}} )
( varnothing )
( {x | x = -3} )

Explanation:

Step1: Analyze the graph of \( y = f(x) \)

The graph is a parabola opening upwards (since the coefficient of \( x^2 \) is positive, as the parabola opens up) with vertex at \( x = -3 \) and \( y = 0 \). For a parabola opening upwards, the function \( f(x) \) is non - negative (\( f(x)\geq0 \)) for all real numbers \( x \) because the minimum value of the function (at the vertex) is \( 0 \) and the function increases as we move away from the vertex in either direction.

Step2: Determine the solution set

Since \( f(x)\geq0 \) for all real numbers \( x \) (because the parabola is above or on the \( x \) - axis for all \( x \) - values), the solution set of the inequality \( f(x)\geq0 \) is all real numbers.

Answer:

\(\{x\mid x\in\mathbb{R}\}\)