If the function $f(x)=2x^3-9ax^2+12a^2x+1, a>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$,then $\alpha$ and $\alpha^2$ are the roots of the equation:

Find two numbers whose sum is $24$ and whose product is as large as possible.

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

If the absolute maximum value of the function $f(x) = (x^2 - 2x + 7) e^{(4x^3 - 12x^2 - 180x + 31)}$ in the interval $[-3, 0]$ is $f(\alpha)$,then:

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$,then $p^{\prime}(0)$ is

The function $f(x) = \int_{-1}^x {t({e^t} - 1)(t - 1){(t - 2)}^3{(t - 3)}^5} dt$ has a local minimum at $x = $ ..........