Question

If $y=f\left(x\right)=a{x}^2+bx+c$ is a quadratic polynomial, then

• A. Its graph is concave upward if $a>0$
• B. Its graph is concave downward if $a<0$
• C. $\frac{d^{2}y}{d{x}^2}=2a$
• D. $\frac{d^{2}y}{d{x}^2}=c$

Answer 1

Answer: (A), (B), (C)

The correct options are
A Its graph is concave upward if $a>0$
B Its graph is concave downward if $a<0$
C $\frac{d^{2}y}{d{x}^2}=2a$

Given, the quadratic polynomial
$y=f\left(x\right)=a{x}^2+bx+c$
Differentiating w.r.t. $x$
$\frac{dy}{dx}=2a{x}^{(2-1)}+b{x}^{(1-1)}+0\Rightarrow \frac{dy}{dx}=2ax+b$

Again, differentiating w.r.t. $x$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=2a{x}^{(1-1)}+0=2ax$

We know that,
If $a>0$ then its graph is concave upward
If $a<0$ then its graph is conave downward.