Concavity

Q.

The which interval the given function $f\left(x\right)=-2x^3-9x^2-12x+1$ is decreasing

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

2. $\left[-2,-1\right]$

3. $\left(-\infty ,-1\right)$

4. $\left(-\infty ,-2\right)$ and $\left(-1,-\infty \right)$

Q.

What is a Derivative in Math?

Q.

What do you mean by derivative?

Q.

$f\left(x\right)=-3x^2+2x+5$ is concave at

1. 3

2. 2

3. 1

4. -2

Q. The value(s) of $a,$ for which the curve $f\left(x\right)=x^4+3a{x}^3+6x^2+5$ is not situated below any of its tangent lines is/are

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

Q.

If a function f(x) is concave at x = a, then f”(a) < 0.

1. True

2. False

Q. If the slope of the tangent to the curve $xy+ax+by=0$ at the point $(1,1)$ on it is $2$, then values of $a$ and $b$ are

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

Q. Suppose the function $f\left(x\right)-f\left(2x\right)$ has the derivative $5$ at $x=1$ and derivative $7$ at $x=2.$ The derivative of the function $f\left(x\right)-f\left(4x\right)$ at $x=1$ has the value equal to

1. $9$
2. $17$
3. $14$
4. $19$

Q. If a tangent of slope $2$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is normal to the circle $x^2+y^2+4x+1=0$, then the maximum value of $ab$ is

1. $4$
2. $2$
3. $1$
4. $Noneofthese$

Q. Let $f\left(x\right)$ be an invertible function such that $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ for all $x\in R,$ then which of the following is/are correct ?
(where $x_1,x_2,\cdots ,x_n$ are different points)

1. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}>f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
2. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}<f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
3. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}<f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
4. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}>f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$

Q. Let $f\left(x\right)$ be an invertible function such that $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ for all $x\in R,$ then which of the following is/are correct ?
(where $x_1,x_2,\cdots ,x_n$ are different points)

1. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}>f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
2. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}<f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
3. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}<f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
4. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}>f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$

Q.

$f\left(x\right)=-3x^2+2x+5$ is concave at

1. 3

2. 2

3. 1

4. -2

Q. The point on the curve $y=$ $3x^2+2x+5$ at which the tangent is perpendicular to the line $x+2y+3=0$.

1. $(0,-5)$
2. $(0,5)$
3. $(-5,0)$
4. $(5,0)$

Q.

$f\left(x\right)=-3x^2+2x+5$ is concave at

1. 3

2. 2

3. 1

4. -2

Q.

If a function f(x) is concave at x = a, then f”(a) < 0.

1. True

2. False

Q. The value(s) of $a,$ for which the curve $f\left(x\right)=x^4+3a{x}^3+6x^2+5$ is not situated below any of its tangent lines is/are

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

Q. The slope of the tangent to the curve $y=-x^3+3x^2+9x-27$ is maximum when x equals.

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

Q.

If a function f(x) is concave at x = a, then f”(a) < 0.

1. True

2. False

Q. At what point of the curve $y=2x^2-x+1$ tangent is parallel to $y=3x+4$

1. $(1,2)$
2. $(0,1)$
3. $(-1,4)$
4. $(2,7)$

Q. If $f\left(x\right)=min\{|x-1|,|x-2|,|x-3|\},$ then the value of $I=\underset{0}{\overset{3}{\int }}f\left(x\right)dx$ equals

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

Q. The curve $y-e^{xy}+x=0$ has a vertical tangent at the point

1. $(1,1)$
2. $nopoint$
3. $(0,1)$
4. $(1,0)$

Q. Let $f\left(x\right)$ be an invertible function such that $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ for all $x\in R,$ then which of the following is/are correct ?
(where $x_1,x_2,\cdots ,x_n$ are different points)

1. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}>f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
2. $\frac{\sum _{r=1}^{n}f\left(x_r\right)}{n}<f\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
3. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}<f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$
4. $\frac{\sum _{r=1}^n{f}^{-1}\left(x_r\right)}{n}>f^{-1}\left(\frac{\sum _{r=1}^n{x}_r}{n}\right)$

Q. The portion of the tangent to xy $=a^2$ at any point on it between the axes is

1. bisected at that point
2. constant
3. Trisected at that point
4. with ratio $1:4$ at the point

Q. The interval of $f\left(x\right)=x^3+2x^2+5x,x<0$ for which it is concave upwards is