Question

$f\left(x\right)={\sin }^{4}x+{\cos }^{4}x+5$, then $f\left(x\right)\le$

• A. $2$
• B. $3$
• C. $6$
• D. $7$

Answer 1

Answer: (C)

$6$

$f\left(x\right)={\sin }^{4}x+{\cos }^{4}x+5$

$=({\sin }^{2}x+{\cos }^{2}x{)}^2-2{\sin }^{2}x{\cos }^{2}x+5$

$=1+5-\frac{(\sin 2x{)}^2}{2}$

$f\left(x\right)=6-\frac{(\sin 2x{)}^2}{2}$

range of $\left(\sin 2x{)}^2ϵ\right[0,1]$

range of $f\left(x\right)ϵ[\frac{11}{2},6]$

$\therefore f\left(x\right)\le 6$