Question

Consider the parabolic function ax^2+bx+c , where a is not equal to 0 , b and c are constants. For what values of a b, and c is f

(i) concave up?

(ii) concave down?

Answer 1

$$f(x)=ax^2+bx+c$$

Domain: $(-\infin, \infin)$

$$f'(x)=2ax+b$$

$$f''(x)=2a$$

(i) If $a>0$ regardless of the values of $b$ and $c,$ $f''(x)>0,$ and $f$ is concave up on $(-\infin, \infin).$

(ii) If $a<0$ regardless of the values of $b$ and $c,$ $f''(x)<0,$ and $f$ is concave down on $(-\infin, \infin).$