Learn Intervals in which the function is increasing or decreasing, Practice increasing and decreasing test & interval of decrease.
\(f(x_2)>f(x_1)\)
whenever \(x_2>x_1\;\;\forall\;x_1,\,x_2\,\in(a,\,b)\).
For example: Observe the given graph.
Here, the function \(y=f(x)\) is increasing in the interval \((b,\,c)\).
In the interval \((b,\,c)\), as the value of \(x\) is increasing in \((b,\,c)\), the value of \(f(x)\) is also increasing.
Here, the function \(y=f(x)\) is increasing in the interval \((b,\,c)\).
In the interval \((b,\,c)\), as the value of \(x\) is increasing in \((b,\,c)\), the value of \(f(x)\) is also increasing.
A \(f\) is increasing in \((a,b) \cup (c,e)\)
B \(f\) is increasing in \((b,c) \cup (d,e)\)
C \(f\) is increasing in \((b,d) \cup (d,e)\)
D \(f\) is increasing in \((a,b) \cup (c,d)\)
Option B is Correct
A \(f\) is increasing in \((-\infty,2) \cup(3,\infty)\) and decreasing in \((2,3)\)
B \(f\) is increasing in \((-\infty,1) \cup(4,\infty)\) and decreasing in \((1,4)\)
C \(f\) is increasing in \((-\infty,5) \cup(6,\infty)\) and decreasing in \((5,6)\)
D \(f\) is increasing in \((-\infty,7) \cup(9,\infty)\) and decreasing in \((7,9)\)
A \(f\) is increasing in \(\left(\dfrac{\pi}{3},\dfrac{5\pi}{3}\right)\) and decreasing in \(\left(0,\dfrac{\pi}{3}\right) \cup \left(\dfrac{5\pi}{3},2\,\pi\right)\)
B \(f\) is increasing in \(\left(0,\pi\right) \cup\left(\dfrac{5\,\pi}{3},2\,\pi\right) \) and decreasing in \(\left(0,\dfrac{\pi}{3}\right) \cup \left(\pi,\dfrac{5\pi}{3}\right)\)
C \(f\) is increasing in \(\left(\dfrac {\pi}{2},\pi\right) \) and decreasing in \(\left(0,\dfrac{\pi}{2}\right) \cup \left(\pi,2\,\pi\right)\)
D \(f\) is increasing in \(\left(\dfrac{\pi}{6},\dfrac{5\pi}{6}\right)\) and decreasing in \(\left(0,\dfrac{\pi}{6}\right) \cup \left(\dfrac{5\pi}{6},2\,\pi\right)\)
For interval of increase \(\to\)\( f' (x) >0\)
For interval of decrease \(\to\)\( f' (x) <0\)
\(f'(x) = 1-2\,cos\,x\)
\(f'(x) > 0 \Rightarrow 1-2\,cos\,x > 0\,\)
\(\Rightarrow cos\,x<\dfrac{1}{2}\)
\(cos\,x= \dfrac{1}{2}\) , when \(x= \dfrac{\pi}{3},\dfrac{5\pi}{3} \)
| Now Interval | \(f'(x)= 1-2\,cos\,x\) | \(f(x)= x-2\,sin\,x\) |
| \(0<x<\dfrac{\pi}{3}\) | \(-\) | Decreasing |
| \(\dfrac{\pi}{3}<x<\dfrac{5\pi}{3}\) | \(+\) | increasing |
| \(\dfrac{5\pi}{3}<x<2\,\pi\) | \(-\) | Decreasing |
\(f\) is increasing in \(\left(\dfrac{\pi}{3},\dfrac{5\,\pi}{3}\right)\)
\(f\) is decreasing in \(\left(0,\dfrac{\pi}{3}\right) \cup \left(\dfrac{5\pi}{3},2\,\pi\right)\)
Option A is Correct
A \(f\) is decreasing in \((a,b) \cup (c,d)\)
B \(f\) is decreasing in \((b,c) \cup (d,e)\)
C \(f\) is decreasing in \((a,c) \cup (d,e)\)
D \(f\) is decreasing in \((b,d) \cup (d,e)\)
Option A is Correct
The slope of the tangent at point \(P=\tan\theta_2\)
The slope of the tangent at point \(Q=\tan\theta_1\)
Here, \(\theta_1\) and \(\theta_2\) are acute angles.
Thus, the slopes, \(\tan\theta_1\) and \(\tan\theta_2\) are positive.
\(f'(x)>0\)
\(f'(x)>0\)
Step 1: Find \(f'(x)\)
\(f'(x)=\dfrac{d}{dx}(2x^2+x+2)\)
\(f'(x)=4x+1\)
Step 2: Apply condition of an increasing function.
\(f'(x)>0\)
\(4x+1>0\)
\(4x>-1\)
\(x>\dfrac{-1}{4}\)
So, the interval for the increase of the function is
\(\left(\dfrac{-1}{4},\;\infty\right)\)
A \(\left(\dfrac{3}{2}, \infty\right)\)
B \(\left(-\infty,\dfrac{5}{4} \right)\)
C \(\left(\dfrac{5}{4},\infty \right)\)
D \(\left(-\infty,-\dfrac{3}{2} \right)\)
A \(\left(-\infty,\dfrac{1}{2} \right)\)
B \(\left(-\dfrac{5}{4},\infty \right)\)
C \(\left(\dfrac{5}{4},\infty \right)\)
D \(\left(\dfrac{3}{2},\infty \right)\)
