To find the values of m for which the line represented by g(x) = mx + 2 becomes a tangent to the graph of f(x) = -3x² + 9x + 2, we need to find the point(s) of tangency. The only value of m for which the line of g(x) becomes a tangent to the graph of f(x) is m = 9.
To find the values of m for which the line represented by g(x) = mx + 2 becomes a tangent to the graph of f(x) = -3x² + 9x + 2, we need to find the point(s) of tangency. A line is tangent to a curve when it touches the curve at exactly one point without crossing through it. Therefore, we need to find the x-coordinate(s) of the point(s) where the graphs of f(x) and g(x) intersect.
To find the x-coordinate(s) of the point(s) of intersection, we equate the expressions for f(x) and g(x) and solve for x:
-3x² + 9x + 2 = mx + 2
This is a quadratic equation. We can rearrange it to the form -3x² + (9 - m)x = 0. For the line to be tangent to the graph, there must be exactly one real root. This means that the discriminant, which is (9 - m)² - 4(-3)(0), must be equal to zero (no real roots) or greater than zero (one real root).
Simplifying the discriminant, we have (9 - m)² = 0 or (9 - m)² > 0.
If (9 - m)² = 0, then 9 - m = 0 and m = 9.
If (9 - m)² > 0, then the line intersects the graph at two distinct points, which means it is not a tangent.
Therefore, the only value of m for which the line of g(x) becomes a tangent to the graph of f(x) is m = 9.