A1. At which points on the curve, given by the equation $f(x)=x^{3}-2 x^{2}+3$, does the tangent make an angle of $135^{\circ}$ with the x-axis?
(A) $T_{1}(1,-4)$ and $T_{2}(-2,0)$.
(B) $T_{1}(1,-4)$ and $T_{2}(2,0)$.
(C) $T_{1}(1,0)$ and $T_{2}(-1,4)$.
(D) $T_{1}(1,2)$ and $T_{2}\left(\frac{1}{3}, \frac{28}{9}\right)$
(E) $T_{1}(1,2)$ and $T_{2}\left(\frac{1}{3}, \frac{76}{27}\right)$.
A1. The value of the derivative of the function $f$ at the sought points must be equal to the tangent of the slope angle of the tangent $\mathrm{v}$ at these points $f^{\prime}(x)=\tan 135^{\circ}=-1$. Solve the equation $3 x^{2}-4 x=-1$. Organize the equation and calculate the abscissas of the sought points $x_{1}=1$ and $x_{2}=\frac{1}{3}$. Calculate the function values $f\left(x_{1}\right)=$ $f(1)=2$ and $f\left(x_{2}\right)=f\left(\frac{1}{3}\right)=\frac{76}{27}$. Write the points $T_{1}(1,2)$ and $T_{2}\left(\frac{1}{3}, \frac{76}{27}\right)$. The correct answer is E.