Maxima and Minima

Maximum value of $f\left(x\right)=4x-\frac{x^2}{2}; x\in \left[-2,\frac{9}{2}\right]$ is

Find the points of local maxima/minima of following function:

$y=\frac{3x^2+4x+4}{x^2+x+1}$

Find the points of local maxima/minima of following function:

$f\left(x\right)=-{\left(x-1\right)}^3{\left(x+1\right)}^2$

The graph of the derivative $f^{\text{'}}$ of a continuous function $f$ is shown with $f\left(0\right)=0$

(i) On what intervals, is $f$ increasing or decreasing?

(ii) At what values of $x$ does $f$ have a local maximum or minimum?

(iii) On what intervals, is $f$ concave upward or downward?

(iv) State the $x$-coordinate(s) of the point(s) of inflection.

(v) Assuming that $f\left(0\right)=0$, sketch a graph of $f$. 

The function $f\left(x\right)$ defined for all real numbers $x$ has the following properties

$f\left(0\right)=0, f\left(2\right)=2$ and $f^{\text{'}}\left(x\right)=k\left(2x-x^2\right)e^{-x}$ for some constant $k>0$. Find

(a) the intervals on which $f$ is increasing and decreasing and any local maximum or minimum values.

(b) the intervals on which the graph $f$ is concave down and concave up

(c) the function $f\left(x\right)$ and plot its graph.

Consider the function, $F\left(x\right)={\int }_{-1}^x\left(t^2-t\right)dt, x\in R$. Find

(a) The $x$ and $y$ intercept of $F$ if they exist.

(b) Derivatives $F\text{'}\left(x\right)$ and $F\text{'}\text{'}\left(x\right)$.

(c) The interval on which $F$ is an increasing and the intervals on which $F$ is decreasing.

(d) Relative maximum and minimum points.

(e) Any inflection point.

Find the points of local maxima/minima of following function

$f\left(x\right)=2x^3-21x^2+36x-20$

What are the dimensions of the rectangular plot of the greatest area which can be laid out within a triangle of base $36 ft$. & altitude $12 ft$? Assume that one side of the rectangular lies on the base of the triangle.

If $b>a$, find the minimum value of $\left|(x-a{)}^3\right|+\left|(x-b{)}^3\right|,x\in R$

A rectangular sheet of poster has its area $18 m^2$. The margin at the top & bottom are $75 cm$ and at the sides $50 cm.$ What are the dimensions of the poster, if the area of the printed space is maximum?

The graph of the derivative $f^{\text{'}}$ of a continuous function $f$ is shown with $f\left(0\right)=0.$ If

(i) $f$ is monotonic increasing in the interval $\left[a, b\right)\cup \left(c, d\right)\cup \left(e, f\right]$ and decreasing in $\left(p, q\right)\cup \left(r, s\right)$.

(ii) $f$ has local minima at $x=x_1$and $x=x_2$.

(iii) $f$ is concave up in $\left(\mathcal{l}, m\right)\cup \left(n, t\right]$.

(iv) $f$ has inflection point at $x=k$.

(v) number of critical points of $y=f\left(x\right)$ is $\text{'}w\text{'}$.

The value of $\left(a+b+c+d+e\right)+\left(p+q+r+s\right)+\left(\mathcal{l}+m+n\right)+\left(x_1+x_2\right)+\left(k+w\right)$ is

The flower bed is to be in the shape of a circular sector of radius $r$ and central angle $\theta$. If the area is fixed & perimeter is minimum, then value of $\theta$ is

Let $x$ and $y$ be two real variable such that $x>0$ and $xy=1$. Find the minimum value of $x+y$.

Investigate the behaviour of the function $y=\left(x^3+4\right)/(x+1{)}^3$ and construct its graph. How many solutions does the equation $\left(x^3+4\right)/(x+1{)}^3=c$ possess ?

Investigate the behaviour of the following function and construct its graph

Investigate the behaviour of the following function and construct its graph

$y=2·e^{x^2-4x}$

Maximum value of $f\left(x\right)=3x^4-8x^3+12x^2-48x+25; x\in \left[0, 3\right]$ is

Investigate the behaviour of the following function and construct its graph

$y=e^x/x$

Investigate the behaviour of the following function and construct its graph

$y=x+\sin x$

Investigate the behaviour of the following function and construct its graph

$y=(x+3)/(x-1)$