The function $f(x) = x^{-x}, (x \in R)$ attains a maximum value at $x =$

The condition that $f(x) = ax^3 + bx^2 + cx + d$ has no extreme value is

The difference between the greatest and the least values of the function $f(x) = x(\ln x - 2)$ on the interval $[1, e^2]$ is:

The function $f(x) = x^3 - 6x^2 + 9x + 2$ has a maximum value when $x$ is

The number of distinct real roots of the equation $x^{7}-7x-2=0$ is

The curvilinear trapezoid is bounded by the curve $y = x^2 + 1$ and the straight lines $x=1$ and $x=2$. The coordinates of the point $(x_1, y_1)$ on the given curve with abscissa $x_1 \in [1, 2]$,where the tangent drawn cuts off an ordinary trapezium of the greatest area from the curvilinear trapezoid,are