Let $\phi(x) = f(x) + f(1-x)$ and $f^{\prime \prime}(x) < 0$ in $[0, 1]$,then
- A$\phi$ is monotonic increasing in $\left[0, \frac{1}{2}\right]$ and monotonic decreasing in $\left[\frac{1}{2}, 1\right]$
- B$\phi$ is monotonic increasing in $\left[\frac{1}{2}, 1\right]$ and monotonic decreasing in $\left[0, \frac{1}{2}\right]$
- C$\phi$ is neither increasing nor decreasing in any sub-interval of $[0, 1]$
- D$\phi$ is increasing in $[0, 1]$
Explore More
Similar Questions
For what values of $K$ is the function $f(x) = Kx^3 + 9x^2 + 9x + 3$ an increasing function on $R$?
Difficult
View SolutionThe set of all $x$ for which $\sin x \leq x$ is
Consider two statements $S_1$ and $S_2$.
$S_1$: If $f(x)$ is a differentiable function with $f'(x) > 0$ in $(a, b)$ and $f(x)$ is increasing in $(a, b)$,then $\frac{f(x)}{f'(x)}$ is also increasing in $(a, b)$.
$S_2$: Both $\sin x$ and $\tan x$ are increasing functions in $(0, \frac{\pi}{2})$.
Which of the following is true?
$S_1$: If $f(x)$ is a differentiable function with $f'(x) > 0$ in $(a, b)$ and $f(x)$ is increasing in $(a, b)$,then $\frac{f(x)}{f'(x)}$ is also increasing in $(a, b)$.
$S_2$: Both $\sin x$ and $\tan x$ are increasing functions in $(0, \frac{\pi}{2})$.
Which of the following is true?
The function $f(x) = \frac{\lambda \sin x + 3 \cos x}{2 \sin x + 6 \cos x}$ is monotonically increasing when:
Difficult
View SolutionFor what values of $x$ is the function $f(x) = x^4 - 4x^3 + 4x^2 + 40$ monotonically decreasing?
Vedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo