Bài tập 35 trang 212 SGK Toán 11 NC

a) Với mọi x ∈ R, ta có:

\(\begin{array}{*{20}{l}}
\begin{array}{l}
y' = 2\cos 2x + 2\sin x\\
 = 2\left( {1 - 2{{\sin }^2}x} \right) + 2\sin x
\end{array}\\
{ =  - 4{{\sin }^2}x + 2\sin x + 2}
\end{array}\)

Vậy \(y\prime  = 0 \Leftrightarrow 2si{n^2}x - sinx - 1 = 0\)

\( \Leftrightarrow \left[ \begin{array}{l}
sinx = 1\\
sinx =  - \frac{1}{2}
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{2} + k2\pi \\
x =  - \frac{\pi }{6} + k2\pi \\
x = \frac{{7\pi }}{6} + k2\pi (k \in Z)
\end{array} \right.\)

b) Với mọi x ∈ R, ta có: y′ = 6cos2x − 8sin2x + 10

Vậy y′ = 0 ⇔ 4sin2x−3cos2x = 5

\( \Leftrightarrow \frac{4}{5}sin2x - \frac{3}{5}cos2x = 1(1)\)

Vì \({\left( {\frac{4}{5}} \right)^2} + {\left( {\frac{3}{5}} \right)^2} = 1\) n

Nên có số α sao cho cosα = 4/5 và sinα = 3/5

Thay vào (1), ta được :

\(\begin{array}{*{20}{l}}
{\sin 2x\cos \alpha  - \sin \alpha \cos 2x = 1}\\
{ \Leftrightarrow \sin (2x - \alpha ) = 1}\\
{ \Leftrightarrow 2x - \alpha  = \frac{\pi }{2} + k2\pi }\\
{ \Leftrightarrow x = \frac{1}{2}\left( {\alpha  + \frac{\pi }{2} + k2\pi } \right)(k \in Z)}
\end{array}\)

c) Với mọi x ∈ R, ta có: y′ = −2cosxsinx + cosx = cosx(1 − 2sinx)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
y\prime  = 0 \Leftrightarrow \cos x(1 - 2\sin x) = 0\\
 \Leftrightarrow \left[ {\begin{array}{*{20}{l}}
{\cos x = 0}\\
{1 - 2\sin x = 0}
\end{array}} \right.
\end{array}\\
{ \Leftrightarrow \left[ {\begin{array}{*{20}{l}}
{x = \frac{\pi }{2} + k\pi }\\
{\sin x = \frac{1}{2} \Leftrightarrow \left[ {\begin{array}{*{20}{l}}
{x = \frac{\pi }{6} + k2\pi }\\
{x = \frac{{5\pi }}{6} + k2\pi }
\end{array}} \right.}
\end{array}} \right.}
\end{array}\)

Vậy \(x = \frac{\pi }{2} + k\pi ;\)

\(x = \frac{\pi }{6} + k2\pi ;x = \frac{{5\pi }}{6} + k2\pi (k \in Z)\)

d)

\(\begin{array}{*{20}{l}}
{y\prime  = \frac{1}{{{{\cos }^2}x}} - \frac{1}{{{{\sin }^2}x}}\forall x \ne k\frac{\pi }{2}}\\
{y\prime  = 0 \Leftrightarrow \frac{1}{{{{\cos }^2}x}} = \frac{1}{{{{\sin }^2}x}} \Leftrightarrow {{\tan }^2}x = 1}\\
{ \Leftrightarrow \tan x =  \pm 1 \Leftrightarrow x =  \pm \frac{\pi }{4} + k\pi ,k \in Z}
\end{array}\)

-- Mod Toán 11 HỌC247