Stationary Point

Q. The point which is equidistant from the points $(-1,1,3),(2,1,2),(0,5,6)$ and $(3,2,2)$ is

1. $(-1,3,4)$
2. $(1,-3,4)$
3. $(1,3,4)$
4. $(4,3,1)$

Q. $\text{If}\underset{x\to 0}{lim}\frac{{64}^x-{32}^x-{16}^x+4^x+2^x-1}{(\sqrt{8+\cos x}-3)\sin x}=a(\ln 2{)}^b,$
then the values of $a$ and $b$ are

1. $a=-36,b=3$
2. $a=-72,b=2$
3. $a=-72,b=3$
4. $a=-36,b=2$

Q. If $△=\left|\begin{array}{ccc}\sin \pi & \cos (x+\frac{\pi }{4})& \tan (x-\frac{\pi }{4})\\ \sin (x-\frac{\pi }{4})& 0& \ln \left(\frac{x}{y}\right)\\ \cot (x+\frac{\pi }{4})& \ln \left(\frac{y}{x}\right)& 0\end{array}\right|,$ for $x\in (0,\pi )-\{\frac{\pi }{4},\frac{3\pi }{4}\},y>0.$ Then the value of $(△+9)=$

Q.

Stationary points are the points where f’(x) is zero or not defined.

1. True

2. False

Q. The sum of all the solution(s) of the equation ${\sin }^{-1}\left(2x\right)={\cos }^{-1}x$ is

1. $0$
2. $\frac{2}{\sqrt{5}}$
3. $\frac{1}{\sqrt{5}}$
4. $-\frac{1}{\sqrt{5}}$

Q. The sum of all possible values of $\theta$ where $\theta \in (0,\frac{\pi }{2}),$ satisfying the equation $\left|\begin{array}{ccc}1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 4\theta \\ {\sin }^2\theta & 1+{\cos }^2\theta & 4\sin 4\theta \\ {\sin }^2\theta & {\cos }^2\theta & 1+4\sin 4\theta \end{array}\right|=0,$ is

1. $\frac{3\pi }{4}$
2. $\frac{5\pi }{4}$
3. $\frac{\pi }{4}$
4. $\frac{11\pi }{12}$

Q. The number of polynomials $p\left(x\right)$ satisfying $\left(p\right(x){)}^2=1+xp(x+1)$ for all real values of $x$ is

1. $0$
2. $1$
3. $2$
4. $4$

Q. The polynomial equation $x^3–3a{x}^2+(27a^2+9)x+2016=0$ has

1. exactly one real root for any real $a$
2. three real roots for any real $a$
3. three real roots for any $a\ge 0$, and exactly one real root for any $a<0$
4. three real roots for any $a\le 0$, and exactly one real root for any $a>0$

Q. $x^x$ has a stationary point at
[Karnataka CET 1993]

1. x = e
2. $x=\frac{1}{e}$
3. $x=\sqrt{e}$
4. x = 1

Q. The curve $f(x,y)=0$ passing through $(0,2)$ satisfy the differential equation $\frac{dy}{dx}=\frac{y^3}{e^x+y^2}.$ If the line $x=\ln 5$ intersects it at points $y=\alpha$ and $y=\beta ,$ then the value of $2|\alpha +\beta |$ is

Q. Find Stationary points of $f\left(x\right)=\sin x$, where $0<x<2\pi$

Q. Find the stationary points of $f\left(x\right)=\cos x$ where $0<x<2\pi$

Q.

A function f(x) is such that it is not differentiable at two points h, k in its domain and f’(x) becomes zero at 3 points a, b, c in the domain. The critical points and stationary points will be -

1. 3, 3

2. 5, 5

3. 5, 3

4. 3, 5

Q. $a-2b+3c$,

Q.

Name the octants in which the following points lie:

(i) (5, 2, 3)

(ii) (-5, 4, 3)

(iii)(4, -3, 5)

(iv) (7, 4, -3)

(v) (-5, -4, 7)

(vi)(-5, -3, -2)

(vii) (2, -5, -7)

(viii)(-7, 2, -5)

Q.
The number of stationary points of $f\left(x\right)=sinx$ in $[0,2\pi ]$ are

1. 1
2. 3
3. 4
4. 2

Q.

If the H.M ofxandyis 2 then show that the points A (x, 2x), B(2y, y) and C(3, 3) are collinear.

What is H.M here ?

Q. $x^x$ has a stationary point at
[Karnataka CET 1993]

1. x = e
2. $x=\frac{1}{e}$
3. x = 1
4. $x=\sqrt{e}$

Q. Suppose that $f$ is a polynomial of degree $3$ and that $f^{\prime \prime }\left(x\right)\ne 0$ at any of the stationary point. Then

1. $f$ has exactly one stationary point
2. $f$ must have no stationary point
3. $f$ must have exactly $2$ stationary point
4. $f$ has exactly $0$ or $2$ stationary point.

Q. $f\left(x\right)=({\sin }^{-1}x{)}^2+({\cos }^{-1}x{)}^2$ is stationary at

1. $x=\frac{1}{\sqrt{2}}$
2. $x=1$
3. $x=\frac{\pi }{4}$
4. $x=0$

Q.

A function f(x) is such that it is not differentiable at two points h, k in its domain and f’(x) becomes zero at 3 points a, b, c in the domain. The critical points and stationary points will be -

1. 5, 5

2. 3, 3

3. 5, 3

4. 3, 5

Q. $\int \frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}dx$ is equal to

1. $\frac{x^2}{2}-x+C$
2. $\frac{x^3}{3}-x+C$
3. $\frac{x^2}{2}+x+C$
4. $\frac{x^2}{2}-2x+C$

Q. The stationary points of $f\left(x\right)=x^3-9x^2+24x-12$ are

Q. The function $y=\frac{ax+b}{(x-1)(x-4)}$ has turning point at $P(2,-1)$. Then find the value of $a$ and $b$.

Q. $x^x$ has a stationary point at
[Karnataka CET 1993]

1. x = e
2. $x=\frac{1}{e}$
3. x = 1
4. $x=\sqrt{e}$

Q.

A function f(x) is such that it is not differentiable at two points h, k in its domain and f’(x) becomes zero at 3 points a, b, c in the domain. The critical points and stationary points will be -

1. 3, 5

2. 3, 3

3. 5, 5

4. 5, 3

Q.

A function f(x) is such that it is not differentiable at two points h, k in its domain and f’(x) becomes zero at 3 points a, b, c in the domain. The critical points and stationary points will be -

1. 3, 3

2. 5, 5

3. 5, 3

4. 3, 5

Q.

Stationary points are the points where f’(x) is zero or not defined.

1. True

2. False

Q. $\Delta \left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \sin 2x+\cos 2x\\ 0& 1& 1\\ 1& 0& -1\end{array}\right|$

1. $(\frac{\pi }{2},\pi )$
2. $(0,\frac{\pi }{4})$
3. $(0,\frac{\pi }{2})$
4. $(-\frac{\pi }{2},0)$

Q. $x^x$ has a stationary point at
[Karnataka CET 1993]

1. x = e
2. $x=\frac{1}{e}$
3. x = 1
4. $x=\sqrt{e}$