Question

If the range of $f\left(x\right)=-x^2+4x-5$ is $(-\infty ,-a)$.Find $a$.

Answer 1

For a parabola $a{x}^2+bx+c$ with vertex $(x_v,y_v)$

If $a<0$ the range is $f\left(x\right)\le y_v$

If $a>0$ the range is $f\left(x\right)\ge y_v$

Here, $a=-1<0$, Vertex $(x_v,y_v)=(2,-1)$

So, $f\left(x\right)\le -1$

$\therefore$ Domain $=(-\infty ,-1)$

Hence, $a=1$