Question

Tìm GTLN, GTNN của:

a, \(f\left(x\right)=3\sin^2x-2\)

b, \(g\left(x\right)=\sin^2x+3\cos x-2\)

Mọi người giúp em với ạ!!! Em cảm ơn nhiều!!!

Answer 1

a/

\(0\le sin^2x\le1\Rightarrow-2\le f\left(x\right)\le1\)

\(f\left(x\right)_{min}=-2\) khi \(sin^2x=1\)

\(f\left(x\right)_{max}=1\) khi \(sin^2x=1\)

b/

\(g\left(x\right)=1-cos^2x+3cosx-2=-cos^2x+3cosx-1\)

\(=-cos^2x+3cosx-2+1=\left(cosx-1\right)\left(2-cosx\right)+1\)

Do \(-1\le cosx\le1\Rightarrow\left\{{}\begin{matrix}cosx-1\le0\\2-cosx>0\end{matrix}\right.\)

\(\Rightarrow\left(cosx-1\right)\left(2-cosx\right)\le0\Rightarrow g\left(x\right)\le1\)

\(g\left(x\right)_{max}=1\) khi \(cosx=1\)

\(g\left(x\right)=-cos^2x+3cosx+4-5=\left(cosx+1\right)\left(4-cosx\right)-5\)

\(\left(cosx+1\right)\left(4-cosx\right)\ge0\Rightarrow g\left(x\right)\ge-5\)

\(g\left(x\right)_{min}=-5\) khi \(cosx=-1\)