Question

O valor minimo da funçao f(x)=x²-kx+15 é -1. o valor de k, sabendo que k<0 é:

a) -10
b) -8
c) -6
d)-1//2
e)-1/8

Answer 1

O valor mínimo da função é dado pelo y do vértice

$Yv=-\dfrac{\Delta}{4a}=f(Xv)\\\\\\Xv=-\dfrac{b}{2a}$
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Calculando o x do vértice:

$X_{v}=-\dfrac{b}{2a}=-\dfrac{-k}{2\cdot1}=\dfrac{k}{2}$

f(Xv) = Yv ----> f(Xv) = -1, logo:

$f(x)=x^{2}-kx+15\\f(Xv)=(Xv)^{2}-k(Xv)+15$

Substituindo Xv:

$f\left(\dfrac{k}{2}\right)=\left(\dfrac{k}{2}\right)^{2}-k\left(\dfrac{k}{2}\right)+15\\\\\\-1=\dfrac{k^{2}}{4}-\dfrac{k^{2}}{2}+15\\\\\\-1-15=\dfrac{k^{2}}{4}-\dfrac{2k^{2}}{4}\\\\\\-16=-\dfrac{k^{2}}{4}\\\\\\k^{2}=16\cdot4\\\\\\k^{2}=64\\\\\\k=\pm\sqrt{64}\\\\\\k=\pm8$

Como k < 0:

$\boxed{\boxed{k=-8}}$