This document provides instructions for verifying Lagrange's theorem using Geogebra. Students are asked to: 1) Open the "Verification of Rolle's theorem" applet and save it with their name. 2) Observe the function f(x)=x^3-3x^2+4 defined on [-1,3] and ensure continuity and differentiability. 3) Record observations of the point A with various slopes and compare the tangent slope to the slope of the chord BC at different intervals. 4) The points that satisfy Lagrange's theorem are where the tangent slope equals the chord BC slope between the endpoints.

1. Math practical-3
Class 12

2. Experiment no. 3
• Aim: Verification of Lagrange’s theorem.
• Software requirements: Geogebra.
• Other requirements: print out of the
graph in the applet, external usb
memory
• Statement: Let f(x) be a real valued
function defined in [a,b] such that
i) f(x) is continuous in [a,b]
ii) f(x) is differentiable in (a,b)
Then Lagrange’s theorem is applicable.
By Lagrange’s theorem, ∃ atleast one c∈
(a,b) such that f’(c) =
𝑓 𝑏 −𝑓(𝑎)
𝑏−𝑎
Draw the figure
here

3. Method followed
1. Open the applet “Verification of Rolle’s theorem” and save
it in the external memory with your name as “Name_
Verification of Rolle’s theorem” .
2. Work on this file now
3. Observe that the function 𝑓 𝑥 = 𝑥3
− 3𝑥2
+ 4 is a real
valued function defined in [-1,3]
4. Ensure that the function h(x) is unselected.
5. By moving the point A, ensure the continuity of the function
f(x)
6. Now select the function h(x), ie. the tangent at A.
7. Move the tangent at the point A to ensure the
differentiability of the function.
8. Measure the slope of BC
8. Find f(a) and f(b)
9. Between [-1,1], record 5 observations of the point A; two
with slope <1, one with slope = 1 and two with slope >1.
10. Repeat the above for the interval [1,3].
S. No. A(x,y) Equation of the
tangent
Slope of the tangent Whether
equal to
Slope of BC
1
2
3
4
5
Observations - 1
Table 1
Observation
S. No. A(x,y) Equation of the tangent Slope of the
tangent
Whether equal to
Slope of BC
1
2
3
4
5
Table 2
Observation
The points x1= and x2= ∈ (−1,3) are the points
where f’(x)= = slope of BC

4. Demonstration
1. The curve is continuous in [-1,3] because___________________
2. The curve is differentiable in the (-3,3) because __________
3. Is f(3)=f(-1)?_____
4. f(a)=_____, f(b)=______
5. Coordinates of the chord joining B and C are ______ and _____
6. Evaluate
𝑓 𝑏 −𝑓(𝑎)
𝑏−𝑎
=____
7. This value is the _______
8. Slope of BC = ______
9. The points that verify Lagrange’s theorem for this function are
_____ and _____ .They ∈ the interval ______
10. At these points, f’(x) = ______ and the tangent at these points is
parallel to _________
11. The value of f(x) at these points is f(x1) = ___ f(x2)=____
Conclusion
Lagrange’s theorem is verified for the points ____ and ____ where the
slope of tangents is equal to the slope of the chord joining the end points
B and C
• Print out here

5. Precautions
1. Copy the file in an external memory
and work on that file, don’t save your
work on the desktop file.
2. Don’t tamper with the in build
functions of the applet ; rather use a
fresh file to make a new applet if
needed.
3. Save the file with a different name in
the external memory, don’t overwrite
on the desktop file.
4. While taking a print out of the graph
applet, change the author’s name to
your own name.
Note :
1. you may download geogebra classic 5 from
https://www.geogebra.org/download
2. You may get a copy of this applet from
https://www.geogebra.org/m/wmd52f4q
3. For all the practicals, visit
https://sites.google.com/view/mathpractical