Practice for finding the Interval of increase and decrease functions from the graph, Identification of local maxima and local minima points of f from the Graph of f?.
\(f(x_2 )> f(x_1) \) whenever \(x_2 >x_1\forall x_1,x_2\varepsilon (a,b)\)
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 \((a,b) \cup (c,d)\)
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 \((b,d) \cup (d,e)\)
2. If \(f' (x)> 0\) in an interval then \(f\) is increasing in the interval.
3. If \(f' (x)< 0\) in an interval then \(f\) is decreasing in the interval.
4. If \(f'' (x)> 0\) then \(f\) is concave upwards .
5. If \(f'' (x)<0\) then \(f\) is concave downwards .
6.\(f(-x) = f(x) \Rightarrow\) Symmetry about \(y\) axis.
1. \(f'>0=f\) is increasing ,so the values of \(x\) for which graph of \(f'\) is above \(x\) axis will be those for which \(f\) increases.
2. \(f'<0=f\) is decreasing , so the values of \(x\) for which graph of \(f'\) is below \(x\) axis will be those for which \(f\)decreases.
A \(f\) is increasing in \((-2,-1) \cup (1,3),\) decreasing in \((-1,1)\).
B \(f\) is increasing in \((-1,3)\) ,decreasing in \((-2,-1)\).
C \(f\) is increasing in \((-2,0)\) decreasing in \((0,3)\).
D \(f\) is increasing in \((-2,1) \cup (2,3),\) decreasing in \((1,2)\).
If graph of \(f'\) goes from above \(x\) axis to below \(x\) axis then that points is local maxima (\(f'\) changes sign positive to negative ) .
A Local maxima at \(x =0\) , Local minima at \(x =2\)
B Local maxima at \(x =-1\) , Local minima at \(x =1\)
C Local maxima at \(x =1\) , Local minima at \(x =-2\)
D Local maxima at \(x =2\) , Local minima at \(x =0\)