The sign of a function and quadratic inequalities
Answers are for parents and teachers.
1
Find the sign of f(x) = 9 − 6x.
Zero at x = 3/2; the coefficient −6 is negative, so f > 0 for x < 3/2 and f < 0 for x > 3/2.
2
Find the sign of f(x) = x² − 4x − 12 and of g(x) = x² − 4x + 9.
f = (x − 6)(x + 2): positive on ]−∞, −2[ ∪ ]6, ∞[, negative on ]−2, 6[. For g: 16 − 36 < 0 and a > 0, so g is positive for every x.
3
Solve x² − 2x − 15 < 0.
(x − 5)(x + 3) < 0: ]−3, 5[.
4
Solve 3x² ≥ 5x + 2.
3x² − 5x − 2 ≥ 0, i.e. (3x + 1)(x − 2) ≥ 0. Roots −1/3 and 2, a > 0: ]−∞, −1/3] ∪ [2, ∞[ (roots included).
5
For which x are f(x) = x + 2 and g(x) = 16 − x² positive together?
f > 0 on ]−2, ∞[ and g > 0 on ]−4, 4[. Together: ]−2, 4[.