If a running track of $500 \ ft$. is to be laid out enclosing a playground,the shape of which is a rectangle with a semicircle at each end,then the length of the rectangular portion such that the area of the rectangular portion is to be maximum is (in feet).

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+ax^2+b \ln|x|+1, x \neq 0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2, -\frac{1}{2}\right]$. Then $|M+m|$ is equal to (Take $\ln 2 \approx 0.7$):

Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x-2) + \alpha x^2 + 4x - \alpha, x > 2$, is

If $x - 2y = 4,$ the minimum value of $xy$ is

If $0 < x < \frac{\pi}{2}$,then the maximum area (in sq. units) of the triangle whose vertices are $(0,0)$,$(x, \cos x)$ and $(\sin^3 x, 0)$ is

The function $f(x)=x^3+a x^2+b x+c$ with $a^2 \leq 3 b$ has: