4Number 4ExplanationGiven the function mx2−2x2+2mx+m−3mx^2 - 2x^2 + 2mx + m - 3mx2−2x2+2mx+m−3. For the function to always be below the xxx-axis, the possible value of mmm is....m<−3m < -3m<−3m<−2m < -2m<−2m<2m < 2m<2m<3m < 3m<3m<115m < 1\frac{1}{5}m<151ExplanationGiven f(x)=mx2−2x2+2mx+m−3f(x) = mx^2 - 2x^2 + 2mx + m - 3f(x)=mx2−2x2+2mx+m−3. Combine like terms to get f(x)=(m−2)x2+2mx+(m−3)f(x) = (m - 2)x^2 + 2mx + (m - 3)f(x)=(m−2)x2+2mx+(m−3) with a=m−2a = m - 2a=m−2, b=2mb = 2mb=2m and c=m−3c = m - 3c=m−3. The condition for the function to always be below the xxx-axis is D<0D < 0D<0 and a<0a < 0a<0. For D<0D < 0D<0 D<0D < 0D<0b2−4ac<0b^2 - 4ac < 0b2−4ac<0(2m)2−4(m−2)(m−3)<0(2m)^2 - 4(m - 2)(m - 3) < 0(2m)2−4(m−2)(m−3)<04m2−4(m2−5m+6)<04m^2 - 4(m^2 - 5m + 6) < 04m2−4(m2−5m+6)<0m2−m2+5m−6<0m^2 - m^2 + 5m - 6 < 0m2−m2+5m−6<05m−6<05m - 6 < 05m−6<05m<65m < 65m<6m<65m < \frac{6}{5}m<56m<115m < 1\frac{1}{5}m<151 For a<0a < 0a<0 a<0a < 0a<0m−2<0m - 2 < 0m−2<0m<2m < 2m<2 The intersection solution of both is m<115m < 1\frac{1}{5}m<151. So, the possible value of mmm below the xxx-axis is m<115m < 1\frac{1}{5}m<151.
