Maxima and Minima

Two particles $P$ and $Q$ located at the point with coordinates $P \left(t, t^3-16t-3\right), Q \left(\mathrm{t+}1, t^3-6t-6\right)$ are moving in a plane. The minimum distance between them in their motion is

$20$ is divided into two parts so that product of cube of one quantity and square of the other quantity is maximum. The parts are

The number that exceeds its square by the greatest amount is

The point for the curve $y=x{e}^x$

From a fixed point $A$ on the circumference of a circle of radius $r$, the perpendicular $AY$ is let fall on the tangent at $P$. The maximum area of the triangle $APY$ is

If the trinomial $x^2+px+q$ has a minimum at $x=3$ and minimum value is equal to $5$, then $p$ and $\mathrm{q}$ are -

The largest term in the sequence $a_n=\frac{n^2}{n^3+200}$ is given by

If $xy =10$ then minimum value of $12x^2 + 13y^2$ is equal to -

The largest value of $2x^3-3x^2-12x+10$ for $-2\le x\le 4$ occurs at $x =$

The minimum distance of the point $\left(a, b, c\right)$ from $x$ -axis is -

If $xy =c^2$, then minimum value of $ax + by$ is -

If $20$ is divided into two parts such that product of one part and the cube of other is maximum, then ratio of the two parts is -

Suppose $f\left(x\right)=\left\{\begin{array}{l}|x-1|+a \mathrm{if} x\le 1\\ 2x+3 \mathrm{if} x>1\end{array}\right.$, if $\mathrm{f}\left(\mathrm{x}\right)$ has a local minimum at $x=1$, then which of the following is most appropriate -

If $y=a\ln \left|x+1\right|+b{\left(x+1\right)}^2+x$ has its extremum value $4$ at $x=0$ , then $\left(a, b\right)$ is -

If $f\left(x\right)=a\log \left|x\right|+b{x}^2+x$ has extreme value at $x =-1$ and $x =2$ then find $a$ and $b$ .

Let $f\left(x\right)=a-{\left(x-3\right)}^{\frac{8}{9}}$, then greatest value of $f\left(x\right)$ is

If $h\left(x\right) =f\left(x\right)+f\left(-x\right)$, then $h\left(x\right)$ has got an extreme value where $f^{\text{'}}\left(x\right)$ is

The greatest value of $\frac{1}{4x^2+2x+1}, x\in R$ is -

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is -

If $0 \le c \le 5$ , then the minimum distance of the point $\left(0, c\right)$ from parabola $y =x^2$ is -