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

Using calculus, to determine concavity, investigate the sign of the second derivative.
<mathjax>#g(x) = -x^2+3x+4#</mathjax>
<mathjax>#g'(x) = -2x+3#</mathjax>
<mathjax>#g''(x) = -2#</mathjax>
Since <mathjax>#g''(x)#</mathjax> is always negative, the graph of <mathjax>#g#</mathjax> is concave down on the domain of <mathjax>#g#</mathjax>. 
Which is to say: <mathjax>#g#</mathjax> is concave down on <mathjax>#(-oo, oo)#</mathjax>.
Using algebra, the graph of this quadratic is a parabola. Since the coefficient of <mathjax>#x^2#</mathjax> is negative, the parabola opens downward. It is concave down.

Using calculus, to determine concavity, investigate the sign of the second derivative.
<mathjax>#g(x) = -x^2+3x+4#
<mathjax>#g'(x) = -2x+3#
<mathjax>#g''(x) = -2#
Since <mathjax>#g''(x)#</mathjax> is always negative, the graph of <mathjax>#g#</mathjax> is concave down on the domain of <mathjax>#g#</mathjax>. 
Which is to say: <mathjax>#g#</mathjax> is concave down on <mathjax>#(-oo, oo)#</mathjax>.
Using algebra, the graph of this quadratic is a parabola. Since the coefficient of <mathjax>#x^2#</mathjax> is negative, the parabola opens downward. It is concave down.

Emma Aldridge

To determine whether the function \( g(x) = -x^2 + 3x + 4 \) is concave up or concave down, you need to find its second derivative. If the second derivative is positive, the function is concave up; if it's negative, the function is concave down. 

First, find the first derivative of \( g(x) \), then find its second derivative. 

\[ g'(x) = -2x + 3 \]

\[ g''(x) = -2 \]

Since the second derivative \( g''(x) \) is a constant (-2), and it's negative, the function is concave down everywhere.

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