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Graphical Interpretation

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?.

 

Intervals in which the Function is Increasing or Decreasing

  • Consider the graph of a function \(f(x)\) as shown. Observe that it is increasing  in the interval (a,b) , decreasing  in (b,c) and again increasing  in (c,d).

  • We say that \(f\) is an increasing function in the interval (a,b) if  

             \(f(x_2 )> f(x_1) \) whenever \(x_2 >x_1\forall x_1,x_2\varepsilon (a,b)\)

 

 

  • We say that \(f\) is a  decreasing function in the interval (a,b) if  \(f(x_2) < f(x_1)\) whenever \(x_2 >x_1\forall x_1,x_2\varepsilon (a,b)\)

 

Illustration Questions

Observe the following graph and indicates the open intervals of increase of function \('f'\).

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 \((a,b) \cup (c,d)\)

×

Observe that \(f\) is decreasing (falling graph) in (a,b) , then increasing (rising graph ) in (b,c) , decreasing  in (c,d) and increasing in (d,e) again.

\(\therefore\) Interval of increase is \((b,c)\cup(d,e)\)

    Interval of decrease is \((a,b)\cup(c,d)\)

 

\(\therefore\) Correct option is (b)

Observe the following graph and indicates the open intervals of increase of function \('f'\).

image
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

Illustration Questions

Observe the following graph and indicates the open interval of decrease of function \('f'\).

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 \((b,d) \cup (d,e)\)

×

Observe that \(f\) is decreasing (falling graph) in (a,b) , then increasing (rising graph) in (b,c) , decreasing  in (c,d) and increasing in (d,e) again.

\(\therefore\) Interval of increase is \((b,c)\cup(d,e)\)

    Interval of decrease is \((a,b)\cup(c,d)\)

 

 

\(\therefore\) Correct option is (A).

Observe the following graph and indicates the open interval of decrease of function \('f'\).

image
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

Concavity of a Curve 

  • If the graph of function  \('f'\)  lies above all the tangents in an interval then it is called concave upwards on that interval.

  • If the graph of function \('f'\) lies below all its tangents  in an  interval , it is called concave downwards in that interval.

Illustration Questions

Which of the following curve is concave upwards in the interval (a,b)?

A

B

C

D

×

Draw tangents to the curve (a,b) at any points  P1,P2, P3. 

Here we can see that the curve lies below the tangent at any given points of contact .So we can say  that the curve is concave downwards .

Thus , this option is incorrect.

 

image

Draw tangents to the curve (a,b) at any points  P1,P2, P3 .

Here, we can see that the curve lies below the tangent for any given point of contact .So we can say that the curve is concave downwards .

Thus , this option is incorrect. 

image

Draw tangents to the curve (a,b) at any points  P1,P2, P3 .

Now  we can see that the curve lies above the tangent for any given points of contact .So ,we can say that the curve is concave upwards  in the  interval (a,b).

Thus , this option is correct .

image

Draw tangents to the curve (a,b) at any points  P1,P2, P3. 

Here, we can see that the curve lies below  the tangent  for any given point of contact .So, we can say the curve  is concave downwards .

Thus , this option is incorrect.

image

Which of the following curve is concave upwards in the interval (a,b)?

A image
B image
C image
D image

Option C is Correct

Sketching the Graph of a Function with Some Given Features 

  • If we desire to sketch  the graph of a function which satisfies certain properties about the derivatives . we make use of the following points .
  1. If \(f' (x)= 0\) then there is a extrema  at that points .

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. 

Illustration Questions

Which of the following is  the sketch of the graph of a function \(f\) which satisfy . 1. \(f'(1) = f'(3) = f'(4) =0 = f(0)\) 2. \(f'(x)>0 \) if \(0<x<1\, {\text or} \,\,\,3<x<4\) 3. \(f'(x)<0\) if \(1<x<3 \,\,{\text or}\,\,\, x>4\) 4. \(f''(x)>0\) if \(0<x<\dfrac{1}{2} \,\,{\text or}\,\,\, x>2\) 5. \(f''(x) <0 \) if  \(\dfrac{1}{2} <x<2\) 6. \(f(-x) = f(x)\)

A

B

C

D

×

\(f(0) = 0\,\Rightarrow\) graph passes through (0,0)

Critical points at \(x = 1,3,4\)

Increasing in \((0,1)\cup(3,4)\) Decreasing  in \((1,3)\cup(4, \infty )\)

Concave upwards in \(\left (0,\dfrac{1}{2}\right) \cup (2,\infty)\), Concave downwards on \(\left (\dfrac{1}{2},2\right)\) 

Symmetric about \(y- axis\) (Draw the graph to the right of \(y- axis\) and take reflection about \(y- axis\))

image

\(\therefore\) Option (a) is correct.

image

Which of the following is  the sketch of the graph of a function \(f\) which satisfy . 1. \(f'(1) = f'(3) = f'(4) =0 = f(0)\) 2. \(f'(x)>0 \) if \(0<x<1\, {\text or} \,\,\,3<x<4\) 3. \(f'(x)<0\) if \(1<x<3 \,\,{\text or}\,\,\, x>4\) 4. \(f''(x)>0\) if \(0<x<\dfrac{1}{2} \,\,{\text or}\,\,\, x>2\) 5. \(f''(x) <0 \) if  \(\dfrac{1}{2} <x<2\) 6. \(f(-x) = f(x)\)

A image
B image
C image
D image

Option A is Correct

Finding the Interval of Increasing and Decreasing from the Graph of  f'(x) 

  • If the  graph of  \(f'\) i.e. derivative of a function \(f\) is given , we can find the interval of  increase or decrease of \(f\) by noting that 

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.

Illustration Questions

The graph of derivative \(f'\)  of a function \(f\) is shown below .Find the intervals in which \(f\) is increasing or decreasing . 

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 \((1,2)\).

×

\(f\) is increasing when \(f' (x)>0\) or graph of \(f'\) is above \(x \to\) axis.

image

\(\therefore f\) increasing in \((-2,-1)\cup(1,3)\)

image

\(f\) decreasing when  \(f'(x) <0\) or graph of \(f'\) is below \(x \to axis\).

image

\(\therefore f\) decreasing in \((-1,1)\)

 

image

The graph of derivative \(f'\)  of a function \(f\) is shown below .Find the intervals in which \(f\) is increasing or decreasing . 

image
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 \((1,2)\).

Option A is Correct

Identification of Local Maxima and Local Minima  Points of f(x) from the Graph of f'(x) 

  • \(f\) has a local maxima and local minima points where \(f'(x)=0\) .So if graph of \(f'\) is given, find the points  where graph of \(f'\) crosses  \(x\)  axis.
  • .These will be the collection of local maxima or minima points .
  • If graph of \(f'\) goes from below \(x\)axis to above \(x\)-axis .then that point is local minima  (\(f'\) changes from negative to positive )

  • 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 ) .  

Illustration Questions

The graph of derivative \(f'\) of a function \(f\) is shown below. Find the local maxima and local minima  points  or the values of \(x \) at which  local maxima and minima  occur .

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 \(x =0\)  

×

Local maxima or minima at the  values  of \(x \) where  \(f'(x) =0\)

So these points  are \(x=-1,1\) (Points where \(f'\) crosses \(x \,\to axis\))

For maxima \(f'\) changes  sign from positive  to negative  i.e. graph of \(f'\) comes below  \(x \to axis\) from above \(x \to axis\)

\(\therefore\, x=-1 \) is a maxima point

For minima \(f'\) changes  sign  from negative to positive  i.e. graph of \(f'\) comes above \(x \to axis\) from below  \(x \to axis\).

\(\therefore x=1\) is a minima point

The graph of derivative \(f'\) of a function \(f\) is shown below. Find the local maxima and local minima  points  or the values of \(x \) at which  local maxima and minima  occur .

image
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 \(x =0\)

 

Option B is Correct

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