How do you determine whether the function $f(x)=x^3+3x^2+5x+7$ is concave up or concave down and its intervals?

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Answer 1 · 2015-07-26T16:54:52

Investigate the sign of the second derivative, $f''(x)$

For $f(x)=x^3+3x^2+5x+7$ , we have

$f'(x)=3x^2+6x+5$ and

$f''(x) = 6x+6$

$f''(x) = 0$ at $x= -1$ , so we check the sign of $f''$ on each side of $-1$

$f''(x)$ is negative (that is $f''(x) < 0$ ) for $x < -1$ . So the graph of $f$ is concave down on $(-oo, -1)$ .

$f''(x) >0$ for $x > -1$ , so the graph of $f$ is concave up on $(-1,oo)$

The point $(-1, 4)$ is the inflection point for the graph.

How do you determine whether the function f(x)=x^3+3x^2+5x+7f(x)=x^3+3x^2+5x+7 is concave up or concave down and its intervals?