Answer
See the sketch.
Work Step by Step
The domain of $f$ is (−∞,∞) and there is no symmetry. There are no asymptotes since f is a polynomial. $f(x) = 3x^2 − 147 = 3(x + 7)(x − 7)$, which is $0$ for $x = 7$ and $x = −7$. $f'(x) > 0$ on (−∞,−7), and on (7,∞), so $f$ is increasing on those intervals. $f'(x) < 0$ on (−7, 7), so $f$ is decreasing on that interval. There is a local maximum at $x = −7$ and a local minimum at $x = 7$. $f''(x) = 6x$, which is $0$ for $x = 0$. $f''(x) > 0$ on (0,∞), so $f$ is concave up on that interval. $f''(x) < 0$ on (−∞, 0), so $f$ is concave down on that interval. There is a point of inflection at $x = 0$. The y-intercept is 286 and the x-intercepts are at −13, 2, and 11.
