Matematické Fórum

stuart clark · 05. 03. 2015 05:24

$(1)$ The smallest positive integer $p$ for which the ex[ression $x^2-2px+3p+4$ is negative for at least one real $x$ is

$(2)$ The real values of $a$ for which the inequality $x^2+ax+a^2+6a<0$ is satisfied for all $x\in \left[1,2\right]$ .

holyduke

↑ stuart clark:
$(1)$
if you derivate this expression, you get $2x-2p$ and when you put that equal to zero, you get $x=p$ . That means, that minimum of this function is in point $p$ .

Now instate this equality to the original expression and you get $-p^{2}+3p+4$ . This expression negative in interval $(-{\infty},-1)$ and $(4,{\infty})$ . That means that the smallest possible integers for p are ${-2}$ and ${5}$ .