Let $\mathrm{f}\left(\mathrm{x}\right) = \mathrm{x} + \frac{1}{\mathrm{x}}$, where x $\in$ (0, 1). Then which one of the following is correct ?

• f(x) fluctuates in the interval

• f(x) increases in the interval

• f(x) decreases in the interval

• None of the above

Which one of the following is correct in respect of the function f(x) = x(x - 1)(x + 1) ?

• The local maximum value is larger than local minimum value

• The local maximum value is smaller than localminimum value

• The function has no local maximum

• The function has no local minimum

Consider the function

$\mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 1\right| + {\mathrm{x}}^2$

where, x $\in \mathrm{ℝ}$

Which one of the following statements is correct ?

• f(X) has local minima at more than one point in $\left( - \infty , \infty \right)$

• f(x) has local maxima at more than one point in $\left( - \infty , \infty \right)$

• f(x) has local minimum at one point only in $\left( - \infty , \infty \right)$

• (x) has neither maxima nor minima in $\left( - \infty , \infty \right)$

Consider the function

$\mathrm{f}\left(\mathrm{x}\right) = {\left(\mathrm{x} - 1\right)}^2\left(\mathrm{x} \hspace{0.17em}+ 1\right){\left(\mathrm{x} - 2\right)}^3$

What is the number of points of local minima of the function f(x) ?

• None

• One

• Two

• Three

Consider the function

$\mathrm{f}\left(\mathrm{x}\right) = {\left(\mathrm{x} - 1\right)}^2\left(\mathrm{x} \hspace{0.17em}+ 1\right){\left(\mathrm{x} - 2\right)}^3$

What is the number of points of local maxima of the function f(x) ?

• None

• one

• Two

• Three

Consider the function

f(x) = $\left|\begin{array}{ccc}{\mathrm{x}}^3& \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ 6& - 1& 0\\ \mathrm{p}& {\mathrm{p}}^2& {\mathrm{p}}^3\end{array}\right|$

where, p is a constant

What is the value of f'(0) ?

• P³

• 3P³

• 6P³

•  - 6P³

Consider the function

f(x) = $\left|\begin{array}{ccc}{\mathrm{x}}^3& \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ 6& - 1& 0\\ \mathrm{p}& {\mathrm{p}}^2& {\mathrm{p}}^3\end{array}\right|$

where, p is a constant

What is the value of P for which f''(0) = 0 ?

• $- \frac{1}{6} \mathrm{or} 0$

•  - 1 or 0

• $- \frac{1}{6} \mathrm{or} 1$

•  - 1 or 1

Which one of the following statements is correct in respect of the function

f(x) = x³sin(x) ?

• It has local maximum at x = 0

• It has local minimum at x = 0

• It has neither maximum nor minimum at x = 0

• It has maximum value 1

19.

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{3{\mathrm{x}}^2 + 12\mathrm{x} - 1, - 1 \le \mathrm{x} \le 2 \\ 37 - \mathrm{x}, 2 < \mathrm{x} \le 3 \right\} \end{gathered}$$

Which of the following statements is/are correct ? 

1. f(x) is increasing in the interval [- 1, 2].
2. f(x) is decreasing in the interval [2, 3]

Select the correct answer using the code given below :

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

What is the value of k for which the sum of the squares of the roots of 2x² – 2(k - 2)x – (k + 1) = 0 is minimum ?

• - 1

• 1

• $\frac{3}{2}$

• 2