Local Minima

Q. Let $f:R\to [1,\infty )$ be a quadratic surjective function such that $f(2+x)=f(2-x)$ and $f\left(1\right)=2.$ Let $g:(-\infty ,\ln 2]\to [1,5]$ be another function defined as $g\left(\ln x\right)=f\left(x\right),$ then which of the following(s) is/are correct ?

1. minimum value of $g^{\prime }\left(x\right)$ is $-2$.
2. $g^{-1}\left(x\right)=\ln (2+\sqrt{x-1})$
3. $g^{-1}\left(x\right)=\ln (2-\sqrt{x-1})$
4. the sum of the values of $x$ that satisfying the equation $f\left(x\right)=5$ is $4.$

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.

Find the value of x if ${|x+1|}^2$ - 5| $|x+1|$ + 6 = 0

1. 1

2. 2

3. -4

4. 3

Q. Find the values of $co{s}^{-1}x$ in terms of given options.

1. $2si{n}^{-1}\sqrt{\frac{1-x}{2}}$
2. $2co{s}^{-1}\sqrt{\frac{1-x}{2}}$
3. $2co{s}^{-1}\sqrt{\frac{1+x}{2}}$
4. $2si{n}^{-1}\sqrt{\frac{1+x}{2}}$

Q. The equation of the conic with focus at (1, −1) directrix along x − y + 1 = 0 and eccentricity $\sqrt{2}$ is
(a) xy = 1
(b) 2xy + 4x − 4y − 1= 0
(c) x² − y² = 1
(d) 2xy − 4x + 4y + 1 = 0

Q. Find the value of other five trigonometric ratios:

Q. The coordinates of the point on the curve $x^3=y(x-a{)}^2,a>0$ where the ordinate is minimum

1. $(2a,8a)$
2. $(-2a,\frac{-8a}{9})$
3. $(3a,\frac{27a}{4})$
4. $(-3a,\frac{-27a}{16})$

Q. Let a function $f:[0,5]\to R$, be continuous, $f\left(1\right)=3$ and $F$ be defined as:
$F\left(x\right)=\underset{1}{\overset{x}{\int }}{t}^{2}g\left(t\right)dt$, where $g\left(t\right)=\underset{1}{\overset{t}{\int }}f\left(u\right)du$. Then for the function $F$, the point $x=1$ is

1. a point of inflection
2. a point of local maxima
3. a point of local minima
4. not a critical point

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