Second Derivative Test Failure Condition

Q.

$x-\left(\frac{1}{x}\right)=3$ then the value of $x^3-\frac{1}{x^3}$ is?

Q.

The minimum value of $x^2+\frac{1}{1+x^2}$ is at

1. $x=0$

2. $x=1$

3. $x=4$

4. $x=3$

Q. Find an anti-derivative (or integral) of $e^{2x}$ by the method of inspection.

Q.

If $f\left(x\right)=4x^3+3x^2+3x+4$, then $x^{3}f\left(\frac{1}{x}\right)$ is equal to:

1. $f\left(-x\right)$

2. $\frac{1}{f\left(x\right)}$

3. ${\left[f\left(\frac{1}{x}\right)\right]}^2$

4. $f\left(x\right)$

Q.

The coefficient of ${\mathrm{x}}^2$ in the expansion of ${\left[\frac{1}{3}{x}^{\frac{1}{2}}+{\left(x\right)}^{\frac{-1}{4}}\right]}^{10}$ is

1. $\frac{70}{243}$

2. $\frac{60}{423}$

3. $\frac{50}{13}$

4. None of these

Q.

If $f\left(x\right)\mathrm{and}g\left(x\right)$ are two functions of $x$ such that $f\left(x\right)+g\left(x\right)=e^x$ and $f\left(x\right)-g\left(x\right)=e^{-x}$, then:

1. $f\left(x\right)$ is odd, $g\left(x\right)$ is odd

2. $f\left(x\right)$ is even, $g\left(x\right)$ is even

3. $f\left(x\right)$ is even, $g\left(x\right)$ is odd

4. $f\left(x\right)$ is odd, $g\left(x\right)$ is even

Q. Find an anti-derivative (or integral) of $(ax+b{)}^2$ by the method of inspection.

Q. Suppose f and g are functions having second derivatives f'' and g'' every where, if f(x).g(x)=1 for all x and f' and g' are never zero, then $\frac{f^{\prime \prime }\left(x\right)}{f^{\prime }\left(x\right)}-\frac{g^{\prime \prime }\left(x\right)}{g^{\prime }\left(x\right)}$ equals

1. 
2. 
3. 
4. 

Q.

In the expansion of $\frac{(e^{7x}+e^{3x})}{e^{5x}}$, the constant term is

1. $0$

2. $1$

3. $2$

4. None of these

Q. Find the anti-derivative $F$ of $f$ defined by $f\left(x\right)=4x^3-6,$ where $F\left(0\right)=3$

Q. At $x=1$, the function $f\left(x\right)=\{\begin{array}{cc}x-1,& x\le 1\\ x^3-1,& x>1\end{array}$ is

1. continuous and differentiable
2. continuous but not differentiable
3. discontinuous and differentiable
4. discontinuous and not differentiable

Q.

Given that f(x) and f’(x) are differentiable at x = c. The Second derivative test will fail if f”(c) is found to be zero

1. False

2. True

Q.

For which of the following functions the second derivative test for finding extremum fails at x = 0 ?

1. f(x) = x²

2. f(x) = x⁴

3. f(x) = x³

4. None of these

Q.

For all real values of x, the minimum value of is

(A) 0 (B) 1

(C) 3 (D)

Q.

For which of the following functions the second derivative test for finding extremum fails at x = 0 ?

1. f(x) = $x^2$

2. f(x) = $x^4$

3. f(x) = $x^3$

4. None of these

Q.

Given that f(x) and f’(x) are differentiable at x = c. The Second derivative test will fail if f”(c) is found to be zero

1. True

2. False

Q.

For which of the following functions the second derivative test for finding extremum fails at x = 0 ?

1. f(x) = $x^2$

2. f(x) = $x^4$

3. f(x) = $x^3$

4. None of these

Q.

Put > or < sign in the boxes:

${(-8)}^2{▭(-6)}^3$

Q.

Put > or < sign in the boxes:

${\left(3\right)}^2\times {\left(4\right)}^2{▭\left(2\right)}^3\times {\left(2\right)}^4$

Q.

Given that f(x) and f’(x) are differentiable at x = c. The Second derivative test will fail if f”(c) is found to be zero

1. True

2. False

Q.

Given that f(x) and f’(x) are differentiable at x = c. The Second derivative test will fail if f”(c) is found to be zero

1. True

2. False

Q.

Find the value of the following: ${\left(3\right)}^0+{\left(4\right)}^0+{\left(9\right)}^0+{\left(16\right)}^0$

Q. $Letf\left(x\right)=x-\left[x\right],x\in R$. $Then{f}^{\prime }\left(\frac{1}{3}\right)=...$

Q.

derrivative of constant iszero. then what is integral zero ? a constant mathematically or zero if viewed practically

Q.

For which of the following functions the second derivative test for finding extremum fails at x = 0 ?

1. f(x) = $x^2$

2. f(x) = $x^4$

3. f(x) = $x^3$

4. None of these

Q.

For which of the following functions the second derivative test for finding extremum fails at x = 0 ?

1. f(x) = x²

2. f(x) = x⁴

3. f(x) = x³

4. None of these