Question:
The function \( f(x) = kx - \sin x \) is strictly increasing for:
The function \( f(x) = kx - \sin x \) is strictly increasing for:
Show Hint
For strict monotonicity, check the sign of the derivative over the entire domain.
Updated On: Jan 13, 2026
- \( k>1 \)
- \( k<1 \)
- \( k>-1 \)
- \( k<-1 \)
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Find the derivative
The derivative of \( f(x) \) is:
\[ f'(x) = k - \cos x. \]
Step 2: Condition for increasing function
For \( f(x) \) to be strictly increasing: \[ f'(x)>0 \implies k - \cos x>0 \implies k>\cos x. \]
Step 3: Maximum value of \( \cos x \)
The maximum value of \( \cos x \) is 1. Therefore: \[ k>1. \]
Step 4: Verify the options
The function is strictly increasing for \( k>1 \), which matches option (A).
The derivative of \( f(x) \) is:
\[ f'(x) = k - \cos x. \]
Step 2: Condition for increasing function
For \( f(x) \) to be strictly increasing: \[ f'(x)>0 \implies k - \cos x>0 \implies k>\cos x. \]
Step 3: Maximum value of \( \cos x \)
The maximum value of \( \cos x \) is 1. Therefore: \[ k>1. \]
Step 4: Verify the options
The function is strictly increasing for \( k>1 \), which matches option (A).
Was this answer helpful?
0
0
Top Questions on Calculus
- A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to :
- The area of the region \[ R=\{(x,y): xy\le 8,\; 1\le y\le x^2,\; x\ge 0\} \] is:
- The area of the region \[ A = \{(x,y) : 4x^2 + y^2 \le 8 \;\text{and}\; y^2 \le 4x\} \] is
If the area of the region \[ \{(x, y) : 1 - 2x \le y \le 4 - x^2,\ x \ge 0,\ y \ge 0\} \] is \[ \frac{\alpha}{\beta}, \] \(\alpha, \beta \in \mathbb{N}\), \(\gcd(\alpha, \beta) = 1\), then the value of \[ (\alpha + \beta) \] is :
- Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3), x \in \mathbb{R}$. Then the value of $f'(5)$ is :
View More Questions
Questions Asked in CBSE CLASS XII exam
- Suppose, the value of Average Propensity to Consume (APC) is 0.8 and National Income is ₹ 4,000 crores, the value of saving would be ₹ ____ crores. (Choose the correct option to fill up the blank)
- CBSE CLASS XII - 2025
- Income and Employment
- Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm²/s. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.
- Two wires of the same material and the same radius have their lengths in the ratio 2:3. They are connected in parallel to a battery which supplies a current of 15 A. Find the current through the wires.
- CBSE CLASS XII - 2025
- Current electricity
- (a) Briefly explain Einstein’s photoelectric equation.
(b) Four metals with their work functions are listed below:
K = 2.3 eV, Na = 2.75 eV, Mo = 4.17 eV, Ni = 5.15 eV.
The radiation of wavelength 330 nm from a laser source placed 1 m away, falls on these metals.
Which of these metals will not show photoelectric emission?
What will happen if the laser source is brought closer to a distance of 50 cm?- CBSE CLASS XII - 2025
- Photoelectric Effect
- Differentiate $2\cos^2 x$ w.r.t. $\cos^2 x$.
- CBSE CLASS XII - 2025
- Derivatives
View More Questions