Here you will learn lagrange’s mean value theorem statement, its geometrical and physical interpretation with examples.
Let’s begin –
Lagrange’s Mean Value Theorem (LMVT)
Statement : Let f be a function that satisfies the following conditions :
(i) f is continuous in [a, b]
(ii) f is differentiable in (a, b)
Then there is a number c in (a, b) such that f'(c) = \(f(b) – f(a)\over b – a\)
Geometrical Interpretation :
Geometrically, the lagrange’s mean value theorem says that somewhere between A and B the curve has atleast on tangent parallel to chord AB.
Physical Interpretation :
If we think of the number (f(b) – f(a))/(b – a) as the average change in f over [a, b] and f'(c) as an instantaneous change, then the mean value theorem says that at some interior point the instantaneous change must equal the average change over the entire interval.
Example : find c of the Lagranges mean value theorem for the function f(x) = \(3x^2\) + 5x + 7 in the interval [1, 3].
Solution : Given f(x) = \(3x^2\) + 5x + 7
\(\implies\) f(1) = 3 + 5 + 7 = 15 and f(3) = 27 + 15 + 7 = 49
Now, Differentiating f(x) with respect to x,
\(\implies\) f'(x) = 6x + 5
Here a = 1, b = 3
Now from lagrange’s mean value theorem
f'(c) = \(f(b) – f(a)\over b – a\) \(\implies\) 6c + 5 = \(49 – 15\over 2\) = 17
\(\implies\) c = 2
Example : If f(x) is continuous and differentiable over [-2, 5] and -4 < f'(x) < 3 for all x in (-2, 5), then the greatest possible value of f(5) – f(-2) is –
Solution : Applying Lagranges mean value theorem (LMVT),
f'(x) = \(f(5) – f(-2)\over 5 -(-2)\) for some x in (-2, 5)
Now, -4 \(\le\) \(f(5) – f(-2)\over 7\) \(\le\) 3
-28 \(\le\) f(5) – f(-2) \(\le\) 21
\(\therefore\) Greatest possible value of f(5) – f(-2) is 21.