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# How do you determine whether the function g(x)=-x^2+3x+4 is concave up or concave down?

Jul 27, 2015

Two methods:

#### Explanation:

Using calculus, to determine concavity, investigate the sign of the second derivative.

$g \left(x\right) = - {x}^{2} + 3 x + 4$

$g ' \left(x\right) = - 2 x + 3$

$g ' ' \left(x\right) = - 2$

Since $g ' ' \left(x\right)$ is always negative, the graph of $g$ is concave down on the domain of $g$.
Which is to say: $g$ is concave down on $\left(- \infty , \infty\right)$.

Using algebra, the graph of this quadratic is a parabola. Since the coefficient of ${x}^{2}$ is negative, the parabola opens downward. It is concave down.