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Find the
local maxima and local minima, if any, of the following functions.
Find also the local maximum and the local minimum values, as the case
may be:

(i). f(x) = x² (ii). g(x)
= x³ − 3x

(iii). h(x)
= sinx + cos, 0 < (iv). f(x)
= sinx − cos x, 0 < x < 2π

(v). f(x) = x³ − 6x²
+ 9x + 15

(vi).

(vii).

(viii).

1. 0.736

If A =$\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]$, find $A^{-1}$. Use it to solve the system of equation 2x-3y+5z = 11, 3x+2y-4z = -5, x+y-2z =-3.

OR

Using elementary row transformations, find the inverse of the matrix

A =$\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$

A straight line through the origin $O$ meets the parallel lines $4x+2y=9$ and $2x+y+6=0$ at $P$ and $Q$ respectively. the point $O$ divides the segment $PQ$ in the ratio

$log\left(sin{1}^{\circ }\right)log\left(sin{2}^{\circ }\right)log\left(sin{3}^{\circ }\right)\dots \dots \dots \dots \dots \dots \dots .log\left(sin{179}^{\circ }\right)$

If $f\left(x\right)=3\sin \sqrt{\frac{{\pi }^2}{16}-x^2},$ then its range is

If $\int \frac{{\sin }^{6}x}{{\cos }^{8}x}dx=\frac{{\left[f\right(x\left)\right]}^{k+1}}{k+1}+C,$ then which of the following is/are true

Half-life of a substance is 20 minutes. What is the time between 33% decay and 67% decay?

The mirror image of $y^2=4x$ about the line $x-y+1=0$ is

Find k, if the straight lines y-3kx+4=0,(2k-1)x-(8k-1) y-6=0

If A and B are any two sets, then $A\cup (A\cap B)$ = ___.

Which of the following curve(s) is(are) symmetric about $y=x$?

For $x>0,$ if $f\left(x\right)={\int }_1^x\frac{{\log }_{e}t}{(1+t)}dt$ then $f\left(e\right)+f\left(\frac{1}{e}\right)$ is equal to:

Find the general solution of cosec x = –2

Findf(x),
where f(x) =

The equation of the tangent to the curve

$y={\sin }^{-1}\frac{2x}{1+x^2}$ at $x=\sqrt{3}$

If $f\left(x\right)=\frac{2-x\cos x}{2+x\cos x}$ and $g\left(x\right)={\log }_{e}x,(x>0)$ then the value of the integral $\underset{-\pi /4}{\overset{\pi /4}{\int }}g\left(f\right(x\left)\right)dx$ is:

Equations x+y=2, 2x+2y=3 will have [UPSEAT 1999]​

A, B have position vectors $\overrightarrow{a},\overrightarrow{b}$ relative to the origin O and X, Y divide $\overrightarrow{AB}$ internally and externally respectively in the ratio 2 : 1. Then,$\overrightarrow{XY}$ is equal to

If $\left|\begin{array}{ccc}a& b+c& a^2\\ b& c+a& b^2\\ c& a+b& c^2\end{array}\right|=0$, where a, b, c are distinct real numbers, then the straight line ax + by + c = 0 passes through the fixed point

The images below are an example for

The eccentricity of the ellipse $a{x}^2+b{y}^2+2fx+2gy+c=0$ if axis of ellipse parallel to x-axis is

If $lo{g}_2(4^{x+1}+4)lo{g}_2(4^x+1)=lo{g}_{2}8,$ then x equals

Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R$, then the line $lx+my+1=0$ touches a fixed circle whose centre and radius of circle are

If $S_n=co{s}^n\theta +si{n}^n\theta$, then the value of $3S_4-2S_6$ is given by

A relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows : $\{x,y\}ϵR\iff x$ is relatively prime to y. Express R as a set of ordered pairs and determine its domain and range.