In the interval $(-4, 4)$,the function $f(x) = \int_{-10}^x (t^4 - 4)e^{-4t} dt$ has:

If $y=a \log x+b x^2+x$ has its extreme value at $x=-1$ and $x=2$,then the value of $a+b$ is

The curve $y(x) = ax^{3} + bx^{2} + cx + 5$ touches the $x$-axis at the point $P(-2, 0)$ and cuts the $y$-axis at the point $Q$,where the derivative $y'(0) = 3$. Find the local maximum value of $y(x)$.

Find all the points of local maxima and local minima of the function $f$ given by $f(x) = 2x^3 - 6x^2 + 6x + 5$.

Let $f:[-1,1] \rightarrow R$ be defined as $f(x)=ax^{2}+bx+c$ for all $x \in[-1,1],$ where $a, b, c \in R$ such that $f(-1)=2, f^{\prime}(-1)=1$ and for $x \in(-1,1)$ the maximum value of $f^{\prime\prime}(x)$ is $\frac{1}{2}.$ If $f(x) \leq \alpha$ for all $x \in[-1,1],$ then the least value of $\alpha$ is equal to:

The maximum area of a rectangle that can be formed with a fixed perimeter of $20$ units is $ . . . . . . $ sq units.