Question

If the range of $f\left(x\right)=x^2+5x-2$ is $(-\frac{m}{4},\infty )$ then the value of $m$ is equal to

Answer 1

$f\left(x\right)=x^2+5x-2$ is a quadratic function with $a=1$ and $b=5$.

Thus, the maximum or minimum value occurs at:

$x=-\frac{b}{2a}=-\frac{5}{2\left(1\right)}=-\frac{5}{2}$

Since $a>0$, the function has the minimum value

$f(-\frac{5}{2})={(-\frac{5}{2})}^2+5(-\frac{5}{2})-2=\frac{25}{4}-\frac{25}{2}-2=\frac{25-50-8}{4}=-\frac{33}{4}$

Hence the range of the given function is $(-\frac{33}{4},+infinity)$.