Second Derivative Test for Local Minimum

Q. The number of distinct real root(s) of $x^4-4x^3+12x^2+x-1=0$ is

Q.

The point $(a^2,a+1)$ lies in the angle between the lines $3x-y+1=0$ and $x+2y-5=0$ contains the origin then

1. $a\in (-3,0)\cup \left(\frac{1}{3},1\right)$

2. $a\in (-\infty ,3)\cup \left(\frac{1}{3},1\right)$

3. $a\in \left(-3,\frac{1}{3}\right)$

4. $a\in \left(\frac{1}{3},\infty \right)$

Q.

If $p\left(x\right)$ be a polynomial of degree three that has a local maximum value $8$ at $x=1$ and a local minimum value $4$ at $x=2$; then $p\left(0\right)$ is equal to

1. $12$

2. $-12$

3. $-24$

4. $6$

Q. Let the function $f$ be defined by $f\left(x\right)=x\ln x$, for all $x>0$. Then

1. $f$ is increasing on $(0,e^{-1})$
2. $f$ is decreasing on $(0,1)$
3. The graph of $f$ is concave up for all $x$
4. The graph of $f$ is concave down for all $x$

Q.

Let $f\left(x\right)$ be a polynomial of degree $6$ in $x$, in which the coefficient of $x^6$ is unity and it has extrema at $x=-1$and $x=1$. If $\underset{x\to 0}{\lim }\left(\frac{f\left(x\right)}{x^3}\right)=1$then$5f\left(2\right)=$

Q.

Given the two vectors $i-j$ and $i+2j,$ the unit vector coplanar with the two perpendicular to that first is$?$

1. $\frac{(i+k)}{\sqrt{2}}$

2. $\frac{(2i+j)}{\sqrt{5}}$

3. $\pm \frac{(i+j)}{\sqrt{2}}$

4. none of these

Q.

Let $f\left(x\right)$ be a polynomial of degree $5$ such that $x=1,-1$ are its critical points. If $\underset{x\to 0}{\lim }\left(2+\frac{f\left(x\right)}{x^3}\right)=4$ then which of the following is not true

1. $f\left(1\right)-4f\left(-1\right)=4$

2. $x=1$ is a point of maxima and $x=-1$ is a point of minimum of $f$

3. $f$ is an odd function.

4. $x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$

Q. If $f"\left(x\right)>0\forall xϵR$ then for any two real numbers $x_{1}and{x}_2,(x_1\ne x_2)$

1. 
2. 
3. 
4. 

Q.

The mid - points of the sides of a triangle are $D(6,1),E(3,5)$ and $F(-1,-2)$, then the vertex opposite to $D$ is

1. $(-4,2)$

2. $(-4,5)$

3. $(2,5)$

4. $(10,8)$

Q. $f\left(x\right)=\{\begin{array}{cc}\frac{\sqrt{(1+px)}-\sqrt{(1-px)}}{x},& -1\le x<0\\ \frac{2x+1}{x-2},& 0\le x\le 1\end{array}$ is continuous in the interval $[-1,1]$, then $p$ is equal to :

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

Q. Let $f\left(x\right)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\underset{x\to 0}{lim}(2+\frac{f\left(x\right)}{x^3})=4$, then which of the following is not true ?

1. $f\left(1\right)-4f(-1)=4$.
2. $x=1$ is a point of maxima and $x=-1$ is a point of minimum of $f$.
3. $f$ is an odd function.
4. $x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$.

Q.

The two parabolas $x^2=4y$ and $y^2=4x$meet in two distinct points. One of these is the origin and the other is

1. $\left(2,2\right)$

2. $\left(4,-4\right)$

3. $\left(4,4\right)$

4. $\left(-2,2\right)$

Q. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Q. For the function f(x) = $x+\frac{1}{x}$

(a) x = 1 is a point of maximum
(b) x = $-$1 is a point of minimum
(c) maximum value > minimum value
(d) maximum value< minimum value

Q. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Q. Show that the right circular cone of least curved surface and given volume has an altitude equal to time the radius of the base.

Q.

Two plots of land have the same perimeter. One is square of side $50m.$while the other is a rectangle whose length is $60m.$Which plot has the greater area and by how much?

Q. $f\left(x\right)$ is a polynomial of the third degree which has a local maximum at $x=-1.$ If $f\left(1\right)=-1,f\left(2\right)=18$ and $f^{\prime }\left(x\right)$ has a local minimum at x=0 then

1. $f\left(0\right)=5$
2. $f\left(x\right)$ has a local minimum at $x=1$
3. $f\left(x\right)$ is increasing in $[1,2\sqrt{5}]$
4. the distance between $(-1,2)$ and $(a,f(a\left)\right),$ where $a$ is a point of local minimum, is $2\sqrt{5}$

Q.

If $\cos \left(x\right)=-\frac{2}{3}$and $x$ is in quadrant III, find the exact value of $\sin \left(2x\right)$, $\cos \left(2x\right)$, and $\tan \left(2x\right)$ algebraically without solving for $x$.

Q. For the given graph of $f\left(x\right)=\cos x.$ Select the correct statements.

1. $g(x{)}_{max}=4\times h(x{)}_{max}$
2. $g\left(x\right)=2\cos x$
3. $h\left(x\right)=0.25\cos x$
4. $h\left(x\right)=0.5+\cos x$

Q.

f(x) and f’(x) are differentiable at x = c. Which of the following is the condition for f(x) to have a local minimum at x = c, if f’(c) = 0

1. None of the above

2. f”(c) < 0

3. f”(c) = 0

4. f”(c) > 0

Q.

The area of a rectangular plot of land of breadth $40m$ is $3000sqm.$ Find its perimeter.

Q. 27. The point on the curve x2- 2y which is nearest to the point (0, 5) is(A) (2V2, 4) (B) (2^2, 0) (C) (o.0) (D) (2, 2)

Q. Let $f\left(x\right)$ be a polynomial function of degree $2$ and $f\left(x\right)>0$ for all $x\in R$. If $g\left(x\right)=f\left(x\right)+f^{{}^{\prime }}\left(x\right)+f^{\prime \prime }\left(x\right)$, then for any $x$

1. $g\left(x\right)<0$
2. $g\left(x\right)=0$
3. $g\left(x\right)>0$
4. $g\left(x\right)\ge 0$

Q. The graph of the function $y=f\left(x\right)$ has unique tangent at the point $(a,0)$ through which the graph passes. Then $\underset{x\to a}{Lt}\frac{\text{log}(1+6f(x\left)\right)}{3f\left(x\right)}$= is

Q. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m 3 . If building of tank costs Rs 70 per sq meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Q. f(x) = x³ $-$ 6x² + 9x + 15

Q. Let h be a twice continuously differentiable positive function on an open interval $J$. Let $g\left(x\right)=ln\left(h\right(x\left)\right)$ for each $xϵJ$
Suppose ${\left(h^{\prime }\left(x\right)\right)}^2>h^{\prime \prime }\left(x\right)h\left(x\right)$ for each $xϵJ$ Then

1. $g$ is increasing on $J$
2. $g$ is decreasing on $J$
3. $g$ is concave up on $J$
4. $g$ is concave down on $J$

Q. If $f\left(x\right)=\left|\begin{array}{ccc}\cos 2x& \cos 2x& \sin 2x\\ -\cos x& \cos x& -\sin x\\ \sin x& \sin x& \cos x\end{array}\right|,$ then

1. $f^{\prime }\left(x\right)=0$ at exactly three points in $(-\pi ,\pi )$
2. $f^{\prime }\left(x\right)=0$ at more than three points in $(-\pi ,\pi )$
3. $f\left(x\right)$ attains its maximum at $x=0$
4. $f\left(x\right)$ attains its minimum at $x=0$

Q. Let $f\left(x\right)$ be a polynomial of degree four having extreme value at $x=1$ and $x=2$. If $\underset{x\to 0}{lim}[1+\frac{f\left(x\right)}{x^2}]=3$, then $f\left(2\right)$ is equal to:

1. $0$
2. $-4$
3. $4$
4. $-8$