JEE Mains · Maths · STD 12 - 6. Application of derivatives
The function \(f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x\)
- A increases in \(\left[\frac{1}{2}, \infty\right)\)
- B increases in \(\left(-\infty, \frac{1}{2}\right]\)
- C decreases in \(\left[\frac{1}{2}, \infty\right)\)
- D decreases in \(\left(-\infty, \frac{1}{2}\right]\)
Answer & Solution
Correct Answer
(A) increases in \(\left[\frac{1}{2}, \infty\right)\)
Step-by-step Solution
Detailed explanation
\(f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x\) \(f^{\prime}(x)=\left(2 x^{2}-x\right)-2 \cos x+2 \cos x-\sin x(2 x-1)\) \(\quad=(2 x-1)(x-\sin x)\) for \(x>0, x-\sin x>0\) \(\quad x<0, x-\sin x<0\) for…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If the function \(f(x)=\left\{\begin{array}{cc}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{array}\right.\) is continuous at \(x=0\), then the value of \(a^2\) is equal toJEE Mains 2024 Hard
- The sum of the distinct real values of \(\mu \), for which the vectors, \(\mu \hat i + \hat j + \hat k,\,\hat i + \mu \hat j + \hat k,\,\hat i + \hat j + \mu \hat k\) are co-planar, isJEE Mains 2019 Medium
- Let the system of linear equations \(x+y+\alpha z=2\) \(3 x+y+z=4\) \(x+2 z=1\) have a unique solution \(\left(x^{*}, y^{*}, z^{*}\right)\). If \(\left(\alpha, x^{*}\right),\left(y^{*}, \alpha\right)\) and \(\left(x^{*},-y^{*}\right)\) are collinear points, then the sum of absolute values of all possible values of \(\alpha\) isJEE Mains 2022 Hard
- Area (in sq. units) of the region outside \(\frac{|\mathrm{x}|}{2}+\frac{|\mathrm{y}|}{3}=1\) and inside the ellipse \(\frac{\mathrm{x}^{2}}{4}+\frac{\mathrm{y}^{2}}{9}=1\) isJEE Mains 2020 Medium
- In the line \( \alpha x+4y=\sqrt{7} \), where \( \alpha\in R \) touches the ellipse \( 3x^{2}+4y^{2}=1 \) at the point P in the first quadrant, then one of the focal distances of P is :JEE Mains 2026 Hard
- The integral \(\int\limits_1^e {\left\{ {\left. {{{\left( {\frac{x}{e}} \right)}^{2x}} - {{\left( {\frac{e}{x}} \right)}^x}} \right\}{{\log }_e}\,x\,dx} \right.} \) is equal toJEE Mains 2019 Hard
More PYQs from JEE Mains
- Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta)=4(\sin^{4}(\frac{7\pi}{2}-\theta)+\sin^{4}(11\pi+\theta)) - 2(\sin^{6}(\frac{3\pi}{2}-\theta)+\sin^{6}(9\pi-\theta)) \), \(\theta \in R\). Then \( \alpha+2\beta \) is equal to:JEE Mains 2026 Medium
- The value of \(3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots \infty}}}}\) is equal toJEE Mains 2021 Medium
- Let \(f(x)=\int \frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)} d x\) . If \(f(3)=\frac{1}{2}\left(\log _e 5-\log _e 6\right)\), then \(f(4)\) is equal toJEE Mains 2023 Medium
- Given below are two statements :
Statement I : \(\lim _{x \rightarrow 0}\left(\frac{\tan ^{-1} x+\log _e \sqrt{\frac{1+x}{1-x}}-2 x}{x^5}\right)=\frac{2}{5}\)
Statement II : \(\lim _{\mathrm{x} \rightarrow 1}\left(\mathrm{x}^{\frac{2}{1-\mathrm{x}}}\right)=\frac{1}{\mathrm{e}^2}\)
In the light of the above statements, choose the correct answer from the options given below :JEE Mains 2025 Medium - Let \(A\) be a \(3 \times 3\) matrix such that \(A^T \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}5\\2\\2\end{bmatrix}\), \(A^T \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}3\\1\\1\end{bmatrix}\), \(A \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}3\\4\\4\end{bmatrix}\) and \(A \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}1\\3\\1\end{bmatrix}\). If \(\det(A) = 1\), then \(\det(\operatorname{adj}(A^2 + A))\) is equal to:JEE Mains 2026 Hard
- An angle between the plane, \(x + y + z = 5\) and the line of intersection of the planes, \(3x + 4y + z- 1 = 0\) and \(5x + 8y + 2z+ 14 = 0\) , isJEE Mains 2018 Hard