可以这样作吗?设|2a+b|=M得$ M^2=4a^2+4ab+b^2 $.由\[ 4a^2-2ab+4b^2-c=0\riff 3b^2-6ab+M^2-c=0\\\Delta =36a^2-12M^2+12c\geqslant 0\riff M^2\leqslant 3a^2+c \]
即|2a+b|取得最大时,$ M^2=3a^2+c $有\[4a^2-2ab+4b^2-M^2+3a^2=0 \\(a-b)^2=0\riff a=b\]有$ 6a^2=c $\[ \dfrac{3}{a}-\dfrac{4}{b}+\dfrac{5}{c}=\dfrac{5}{6a^2}-\dfrac{1}{a} \]
由导数解得$ a=\dfrac{\sqrt{15}}{15} $时最小值为$ \dfrac{75}{6}-\sqrt{15} $ |