Local Minima

Q. The least value of $\alpha ϵR$ for which $4\alpha x^2+\frac{1}{x}\ge 1$, for all x > 0, is

1. $\frac{1}{64}$
2. $\frac{1}{32}$
3. $\frac{1}{27}$
4. $\frac{1}{25}$

Q.

If, for a positive integer n, the quadratic equation, $\mathrm{x}(\mathrm{x}+1)+(\mathrm{x}+1)(\mathrm{x}+2)+....+(\mathrm{x}+\overline{\mathrm{n}-1})(\mathrm{x}+\mathrm{n})=10\mathrm{n}$ has two consecutive integral solutions, then n is equal to:

1. 9

2. 10

3. 11

4. 12

Q.

If $f\left(x\right)=\frac{5x+3}{6x+a}$ and $f\left(f\left(x\right)\right)=x$ then the value of $a$ is

1. $-5$

2. $5$

3. $6$

4. $-6$

Q.

Two towns $A$ and $B$ are $60km$ apart. A school is to be built to serve $150$ students in town $A$ and $50$ students in town $B$. If the total distance to be travelled by all $200$ students is to be as small as possible, then the school should be built at

1. town $B$

2. $45km$ from town $A$

3. town $A$

4. $45km$ from town $B$

Q. Let $f\left(x\right)$ be a polynomial function of degree $4,$ vanishes at $x=-1$. If $f\left(x\right)$ has local maxima/minima at $x=1,2,3$ and $\underset{2}{\overset{-2}{\int }}f\left(x\right)dx=\frac{1348}{15}.$ Then

1. the $y$-intercept of the tangent to the curve at $x=-1$ equals $-96$
2. the maximum value of $f\left(x\right)$ is $-63$
3. $\underset{-1}{\overset{1}{\int }}[f\left(x\right)+f(-x)]dx=-\frac{1424}{15}$
4. the minimum value of $f\left(x\right)$ is $-64$

Q.

If $co{s}^{-1}\frac{3}{5}-si{n}^{-1}\frac{4}{5}=co{s}^{-1}x$ then x equal to

1. 0
2. 1
3. -1
4. None of these

Q.

If the function $f\left(x\right)=2x^3-9a{x}^2+12a^{2}x+1,$ where a > 0, attains its maximum and minimum at p and q respectively such that $p^2=q$, then a equals

1. 3

2. 1

3. 2

4. $\frac{1}{2}$

Q. Let $f\left(x\right)=|x^2-4x+3|$ be a function defined on $x\in [0,4]$ and $\alpha ,\beta ,\gamma$ are the abscissas of the critical points of $f\left(x\right).$ If $m$ and $M$ are the local and absolute maximum values of $f\left(x\right)$ respectively, then the value of ${\alpha }^2+{\beta }^2+{\gamma }^2+m^2+M^2$ is

Q. If $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$ in $(0,\frac{\pi }{2})$, then which of the following options is INCORRECT ?

1. $f$ is strictly increasing in $(\frac{\pi }{4},\frac{\pi }{2})$
2. $f$ is strictly decreasing in $(0,\frac{\pi }{4})$
3. $x=\frac{\pi }{4}$ is a point of local minima
4. $x=\frac{\pi }{4}$ is a point of local maxima

Q. Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable, $f\left(0\right)=f\left(1\right)=0$ and satisfies $f^{\prime \prime }\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$
If the function $e^{-x}f\left(x\right)$ assumes its minimum in the interval [0, 1] at $=\frac{1}{4}$, which of the following is true?

1. $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{3}{4}$
2. $f^{\prime }\left(x\right)<f\left(x\right),\frac{1}{4}<x<\frac{1}{4}$
3. $f^{\prime }\left(x\right)<f\left(x\right),0<x<\frac{1}{4}$
4. $f^{\prime }\left(x\right)<f\left(x\right),\frac{3}{4}<x<1$

Q. Minimum distance between the curves $y^2=x-1$ and $x^2=y-1$ is $\frac{a\sqrt{2}}{b}$, where $a,b$ are co-prime numbers, then the value of $a+b$ is

Q.

The least value of 'a' for which $\frac{4}{sinx}+\frac{1}{1-sinx}=a$ has at least one solution in the interval $(0,\pi /2)$ is

1. 9

2. 8

3. 4

4. 1

Q.

$f\left(x\right)=\sin x\text{has a local minima at}x=\frac{3\pi }{2}$.

1. True

2. False