$(I)$ The curve $y=f(x)$ intersects the $x$-axis exactly at one point
$(II)$ The curve $y=f(x)$ intersects the $x$-axis at $\mathrm{x}=\cos \frac{\pi}{12}$
Then
$\mathrm{f}\left(\frac{1}{2}\right)<0$
$\mathrm{f}(1)>0 \Rightarrow(\mathrm{A})$ is correct.
$f(x)=\sqrt{2}\left(4 x^3-3 x\right)-1=0$
Let $\cos \alpha=\mathrm{x}$,
$\cos 3 \alpha=\cos \frac{\pi}{4} \Rightarrow \alpha=\frac{\pi}{12}$
$\mathrm{x}=\cos \frac{\pi}{12}$
$(4)$ is correct.
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$(\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0$
$(\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0$
$2 x+(3 \lambda+1) y+3(\lambda-1) z=0$
has non-zero solutions, is
$f\left( {x + a} \right) = \frac{1}{2} + \sqrt {f\left( x \right) - {f^2}\left( x \right)}$ a is a real constant, then $f(x)$ must be
$f(0)=g(0)=0$
$\Psi_1( x )= e ^{- x }+ x , \quad x \geq 0$
$\Psi_2( x )= x ^2-2 x -2 e ^{- x }+2, x \geq 0$
$f( x )=\int_{- x }^{ x }\left(| t |- t ^2\right) e ^{- t ^2} dt , x >0$
and
$g(x)=\int_0^{x^2} \sqrt{t} e^{-t} d t, x>0$
($1$) Which of the following statements is $TRUE$ ?
$(A)$ $f(\sqrt{\ln 3})+ g (\sqrt{\ln 3})=\frac{1}{3}$
$(B)$ For every $x>1$, there exists an $\alpha \in(1, x)$ such that $\psi_1(x)=1+\alpha x$
$(C)$ For every $x>0$, there exists a $\beta \in(0, x)$ such that $\psi_2(x)=2 x\left(\psi_1(\beta)-1\right)$
$(D)$ $f$ is an increasing function on the interval $\left[0, \frac{3}{2}\right]$
($2$) Which of the following statements is $TRUE$ ?
$(A)$ $\psi_1$ (x) $\leq 1$, for all $x>0$
$(B)$ $\psi_2(x) \leq 0$, for all $x>0$
$(C)$ $f( x ) \geq 1- e ^{- x ^2}-\frac{2}{3} x ^3+\frac{2}{5} x ^5$, for all $x \in\left(0, \frac{1}{2}\right)$
$(D)$ $g(x) \leq \frac{2}{3} x^3-\frac{2}{5} x^5+\frac{1}{7} x^7$, for all $x \in\left(0, \frac{1}{2}\right)$