21.

Consider the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 1\right| + {\mathrm{x}}^2 \\ \mathrm{where}, \mathrm{x} \in \mathrm{R} \end{gathered}$$

Which one of the following statements is correct ?

• f(x) is continuous but not differentiable at x = 0

• f(x) is continuous but not differentiable at x = 1

• f(x) is differentiable at x = 1

• f(x) is not differentiable at x = 0 and x = 1

Consider the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 1\right| + {\mathrm{x}}^2 \\ \mathrm{where}, \mathrm{x} \in \mathrm{R} \end{gathered}$$

Which one of the following statements is correct ?

• f(x) is increasing in $\left( - \infty , \frac{1}{2}\right)$ and decreasing in $\left(\frac{1}{2}, \infty \right)$

• f(x) is decreasing in$\left( - \infty , \frac{1}{2}\right)$ and increasing in $\left(\frac{1}{2}, \infty \right)$

• f(x) is increasing in $\left( - \infty , 1\right)$ and decreasing in $\left(1, \infty \right)$

• f(x) is decreasing in $\left( - \infty , 1\right)$ and increasing in $\left(1, \infty \right)$

Consider the function

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

where, x $\in \mathrm{ℝ}$

What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates x = 1 and x = $\frac{3}{2}$ ?

• $\frac{5}{12}$ square unit

• $\frac{7}{12}$ square unit

• $\frac{2}{3}$ square unit

• $\frac{11}{12}$ square unit

Consider the equation $\mathrm{x} + \left|\mathrm{y}\right| = 2\mathrm{y}$

Which of the following statements are not correct ?

1. y as a function of x is not defined for all real x.
2. y as a function of x is not continuous at x = 0.
3. y as a function of x is differentiable for all x.

Select the correct answer using the code given below

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

Consider the equation $\mathrm{x} + \left|\mathrm{y}\right| = 2\mathrm{y}$

What is the derivative of y as a function of x with respect to x for x $< 0$ ?

• 2

• 1

• $\frac{1}{2}$

• $\frac{1}{3}$

A function f(x) is defined as follows

$$\begin{gathered} f\left(x\right) = \left\{\mathrm{x} + \mathrm{\pi } \mathrm{for} \mathrm{x} \in [ - \mathrm{\pi }, 0) \\ \mathrm{\pi cos}\left(\mathrm{x}\right) \mathrm{for} \mathrm{x} \in \left[0, \frac{\mathrm{\pi }}{2}\right] \\ {\left(\mathrm{x} - \frac{\mathrm{\pi }}{2}\right)}^2 \mathrm{for} \mathrm{x} \in (\frac{\mathrm{\pi }}{2}, \mathrm{\pi }]\right\} \end{gathered}$$

Consider the following statements :

1. The function f(x) is continuous at x = 0.
2. The function f(x) is continuous at x = $\frac{\mathrm{\pi }}{2}$.

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2 

• Neither 1 nor 2

A function f(x) is defined as follows

$$\begin{gathered} f\left(x\right) = \left\{\mathrm{x} + \mathrm{\pi } \mathrm{for} \mathrm{x} \in [ - \mathrm{\pi }, 0) \\ \mathrm{\pi cos}\left(\mathrm{x}\right) \mathrm{for} \mathrm{x} \in \left[0, \frac{\mathrm{\pi }}{2}\right] \\ {\left(\mathrm{x} - \frac{\mathrm{\pi }}{2}\right)}^2 \mathrm{for} \mathrm{x} \in (\frac{\mathrm{\pi }}{2}, \mathrm{\pi }]\right\} \end{gathered}$$

Consider the following statements :

1. The function f(x) is differentiable at x = 0.
2. The function f(x) is differentiable at x = $\frac{\mathrm{\pi }}{2}$.

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2 

• Neither 1 nor 2

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \mathrm{ℝ} \to \mathrm{ℝ} \mathrm{be} \mathrm{a} \mathrm{function} \mathrm{such} \mathrm{that} \\ \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3 + {\mathrm{x}}^2\mathrm{f}\text{'}\left(1\right) + \mathrm{xf}\text{'}\text{'}\left(2\right) + \mathrm{f}\text{'}\text{'}\text{'}\left(3\right) \\ \mathrm{for} \mathrm{x} \in \mathrm{ℝ} \end{gathered}$$

What is f(1) = ?

•  - 2

•  - 1

• 0

• 4

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \mathrm{ℝ} \to \mathrm{ℝ} \mathrm{be} \mathrm{a} \mathrm{function} \mathrm{such} \mathrm{that} \\ \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3 + {\mathrm{x}}^2\mathrm{f}\text{'}\left(1\right) + \mathrm{xf}\text{'}\text{'}\left(2\right) + \mathrm{f}\text{'}\text{'}\text{'}\left(3\right) \\ \mathrm{for} \mathrm{x} \in \mathrm{ℝ} \end{gathered}$$

What is f'(1) = ?

•  - 6

•  - 5

• 1

• 0

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \mathrm{ℝ} \to \mathrm{ℝ} \mathrm{be} \mathrm{a} \mathrm{function} \mathrm{such} \mathrm{that} \\ \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3 + {\mathrm{x}}^2\mathrm{f}\text{'}\left(1\right) + \mathrm{xf}\text{'}\text{'}\left(2\right) + \mathrm{f}\text{'}\text{'}\text{'}\left(3\right) \\ \mathrm{for} \mathrm{x} \in \mathrm{ℝ} \end{gathered}$$

What is f'''(10) = ?

• 1

• 5

• 6

• 8