Application of Derivatives | SCRA

  
Increasing & Decreasing Functions

1. Let f : be a real function, then
(i) If , then f is said to be increasing in A.
(ii) If , then f is said to be decreasing in A.
(iii) If , then f is said to be strictly increasing in A.
(iv) If , then f is said to be strictly decreasing in A.
(v) A function f(x) is said to be monotonic in A if f(x) is either increasing (or) decreasing in A.

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(i) If f '(x 0 x (a, b) and points which make f '(x) equal to zero (in between (a, b)) don't form an interval, then f(x) would be increasing in [a, b]
(ii) If f '(x 0 x (a, b) and points which make f '(x) equal to zero (in between (a, b)) don't form an interval, f(x) would be decreasing in [a, b]
(iii) If f(0) = 0 and f '(x) 0 x R, then f(x) 0 x ( , 0) and f(x) 0 x (0, )
(iv) If f(0) = 0 and f '(x) 0 x R, then f(x) 0 x ( , 0) and f(x) 0 x (0, )
(v) A function is said to be monotonic if it's either increasing or decreasing.
(vi) The points for which f '(x) is equal to zero or doesn't exist are called critical points. Here it should also be noted that critical points are the interior points of an interval.
(vii) The stationary points are the points where f '(x) = 0 in the domain.
 
 Local Maximum and Local Minimum:
Let y = f(x) be a function defined at x = a and also in the vicinity of the point x = a. Then f(x) is said to have a local maximum at x = a, if the value of the function at x = a is greater than the value of the function at the neighbouring points of x = a. Mathematically, f(a) > f(a h) and f(a) > f(a + h) where h > 0
Similarly, f(x) is said to have a local minimum at x = a, if the value of the function at x = a is less than the value of the function at the neighbouring points of x = a.
Mathematically, f(a) < f(a h) and f(a) < f(a + h) where h > 0
A local maximum or a local minimum is also called a local extremum.


Test for Local Maximum / Minimum
We have two cases to consider

Test for Local Maximum / Minimum at x = a if f(x) is Differentiable at x = a:
If f(x) is differentiable at x = a and if it is a critical point of the function (i.e., f '(a) = 0) then we have the following three tests to decide whether f(x) has a local maximum or local mininim or neither at x = a.

i First Derivative Test:
If f '(a) = 0 and f '(x) changes it's sign while passing through the point x = a, then
(a) f(x) would have a local maximum at x = a if f '(a 0) > 0 and f '(a + 0) < 0. It means that f '(x) should change its sign from positive to negative.
(b) f(x) would have local minimum at x = a if f '(a 0) < 0 and f '(a + 0) > 0. It means that f '(x) should change its sign from negative to positive.
(c) If f(x) doesn't change its sign while passing through x = a, then f(x) would have neither a maximum nor minimum at x = a

ii. Second Derivative Test:
This test is basically the mathematical representatin of the first derivative test. It simply says that,
(a) If f '(a) = 0 and f ''(a) < 0, then f(x) would have a local maximum at x = a.
(b) If f '(a) = 0 and f ''(a) > 0, then f(x) would have a local minimum at x = a.
(c) If f '(a) = 0 and f ''(a) = 0, then this test fails and the existence of a local maximum / minimum at x = a is decided on the basis of the nth derivative test.

iii. n^th Derivative Test:
It is nothing but the general version of the second derivative test. It says that if, f '(a) = f ''(a) = ....... = f ⁿ(a) = 0 and f ^{n + 1}(a) ¹ 0 (all derivatives of the function up to order `n' vanishes and (n + 1)th order derivative does not vanish at x = a, then f(x) would have a local maximum or minimum at x = a iff n is odd natural number and that x = a would be a point of local maxima if f ^{n + 1} (a) < 0 and would be a point of local minima if f ^{n + 1}(a) > 0.
It is clear that the last two tests are basically the mathematical representation of the first derivative test. But that shouldn't diminish the importance of these tests. Because at time it's becomes very difficult to decide whether f '(x) changes it's sign or not while passing through point x = a, and then remaining tests may come handy in these type of situations.

Test for Local Maximum / Minimum at x = a if f(x) is not Differentiable at x = a:
Case I. When f(x) is continuous at x = a and f '(a h) and f '(a + h) exist and are non-zero, then f(x) has a local maximum or minimum at x = a if f '(a h) and f '(a + h) are of opposite signs.
If f '(a h) > 0 and f '(a + h) < 0 then x = a will be a point of local maximum
If f '(a h) < 0 and f '(a + h) > 0 then x = a will be a point of local minimum

Case II. When f(x) is continuous and f '(a h) and f '(a + h) exist but one of them is zero, we should infer the information about the existence of local maxima / minima from the basic definition of local maxima / minima.

Case III. If f(x) is not continuous at x = a and f '(a h) and / or f '(a + h) are not finite, then compare the values of f(x) at the neighbouring point of x = a.
Remarks: It is advisable to draw the graph of the function in the vicinity of the point x = a becasue the graph would give us the clear cicture about the existence of local maxima / minima at x = a.
Global Maximum/Minimum
Let y = f (x) be a given function with domain D. Let [a, b] D. Global maximum/minimum of f(x) in [a, b] is basically the greatest/least value of f(x) in [a, b].
Global maximum and minimum in [a, b] would always occur at critical points of f(x) within [a, b] or at the end points of the interval.
 Global Maximum/Minimum in [a, b]
In order to find the global maximum and minimum of f(x) in [a, b], find out all the critical points of f(x) in (a, b). Let c₁, c₂, ....... c_n be the different cricitial points. Find the value of the function at these critical points. Let f(c₁), f(c₂), ....... f(c_n) be the values of the function at critical points.
Say, M₁ = max{f(a), f(c₁), f(c₂), ..........., f(c_n), f(b)}
and M₂ = min{f(a), f(c₁), f(c₂), ..........., f(c_n), f(b)}
Then M₁ is the greatest value of f(x) in [a, b] and M₂ is the least value of f(x) in [a, t]
Global Maximum/Minimum in (a, b)
Method for obtaining the greatest and least values of f(x) in (a, b) is almost same as the method used for obtaining the greatest and least values in [a, b], however with a caution.
Let y = f(x) be a function and c₁, c₂, ....... c_n be the different critical points of the function in (a, b)
Let M₁ = max {f(c₁), f(c₂), f(c₃) ........... f(c_n)}
and M₂ = min {f(c₁), f(c₂), f(c₃) ........... f(c_n)}
Now if would not have global maximum (or global minimum) in (a, b)
This means that if the limiting values at the end points are greater than M₁ or less than M₂, then f(x) would not have global maximum/minimum in (a, b). On the other hand if M₁ > , then M₁ and M₂ would respectively be the global maximum and global minimum of f(x) in (a, b).

Rolle's Theorem
It is one of the most fundamental theorem of differential calculus and has far reaching consequences. It states that if y = f(x) be a given function and satisfies.
. f(x) is continuous in [a, b]
. f(x) is differentiable in (a, b)
. f(a) = f(b)
Then f '(x) = 0 at least once for some x (a, b)

If f(x) satisfies the conditions of Rolle's theorem in [a, b], it's derivative would vanish at least once in (a, b).
A (a, f(a)), B (b, f(b))
as f(a) = f(b) (third condition of Rolle's theorem)
Slope of line AB = 0
We would have at least one point belonging to (a, b) so that tangent drawn to the curve at that point would be parallel to the line AB.

Illustration: Let f(x) = x² 3x + 4. Test for Rolle's Theorem in [1, 2]
Sol. As f(1) = f(2) = 2
Now, f '(x) = 0 2x- 3 = 0
x = x (1, 2)

Lagrange's Mean Value Theorem
This theorem is in fact the general version of Rolle's theorem. It says that if y = f(x) be a given function which is
. Continuous in [a, b]
. Differentiable in (a, b)
Then
f '(x) = , at least once forsome x (a, b)
Let A (a, f(a)) and B (b, f(a))
Slope of Chord AB =
Application of Derivative in Determining the Nature of Roots of Cubic Polynomial
Let f(x) = x³ + ax² + bx + c be the given cubic polynomial, and f(x) = 0 be the corresponding cubic equation, where a, b, c R.
Now, f '(x) = 3x² + 2ax + b
Let D = 4a² 12b = 4(a² 3b) be the discriminant of the equation f '(x) = 0
(i) If D < 0 f '(x) > 0 x R. That means f(x) would be an increasing function of x. Also f() = , thus the graph of f(x) would look like,

It is clear that graph of y = f(x) would cut the x-axis only once. That means we would have just one real root, (say x₀). Clearly x₀ > 0 if c > 0.
(ii) If D > 0, f '(x) = 0 would have two real roots (say x₁ and x₂, let x₁ < x₂)
f '(x) = 3 (x x₁) (x x₂)
Sign of f '(x)

Clearly, f '(x) < 0, x (x₁, x₂) and f '(x) > 0, x ( , x₁) – (x₂, )
That means f(x) would increase in ( , x₁) and (x₂, ) and would decrease in (x₁, x₂). Hence x = x₁ would be a point of local maxima and x = x₂ would be a point of local minima.

Thus the graph of y = f(x) could have these five possibilities:
(a)
(b)
(c)
(d)
(e)

Clearly in fig. (a) we have three real and distinct roots. In figure (b) and (c) we have just one real root and in figures (d) and (e) we have 3 real roots but one of them would be repeated.
. If f(x₁) f(x₂) > 0, f(x) = 0 would have just one real root.
. If f(x₁) f(x₂) < 0, f(x) = 0 would have three real and distinct roots.
. If f(x₁) f(x₂) = 0, f(x) = 0 would have three real roots but one of the root would be repeated.
(iii) If D = 0, f '(x) = 3 (x x₁)² where x₁ is the root of f '(x) = 0
f(x) = (x x₁)³ + k
Now if k = 0, then f(x) = 0 has three equal real roots and if k¹ 0 then f(x) = 0 has one real root.

Illustration: Find all possible values of the parameter `a' so that x³ 3x + a = 0 has three real and distinct roots.
Sol. Let f(x) = x³ 3x + a
f '(x) = 3x² 3
= 3(x 1) (x + 1)
Clearly x = 1 is the point of maxima and x = 1 is the point of minima.
Now, f(1) = a 2, f( 1) = a + 2

The roots of f(x) = 0 would be real and distinct if
f(1) f( 1) < 0
(a 2) (a + 2) < 0
2 < a < 2
Thus given equation would have real and distinct roots if a ( 2, 2)

Tangent & Normals
Let y = f(x) be a continous function defined over an interval I
1. The slope of the tangent to the curve y = f(x) at the point is and is denoted by m
2. The slope of the tangent to the curve at a point is called the gradient to the curve at that point.
3. Equation of the tangent to the curve y = f(x) at is y - y₁ = m (x - x₁)
4. If m = 0 then the tangent at that point is parallel to x -axis.
5. If m is not defined then the tangent is perpendicular to x-axis
6. A line which is perpendicular to the tangent of the curve at P and passes through P is called the normal to the curve at P.
7. Equation of the normal to the curve y = f(x) at is x - x₁= -m( y - y₁)
 
 Angle of intersection of two plane curves:
1. The angle between two curves is defined as the angle between the two tangents at their common point of intersection.
2. Let y = f₁(x) and y = f₂(x) be two plane curves intersect at P(x₁ , y₁). If m₁ = and m₂ = are the slopes of the tangents at P and be the acute angle between them then tan
3. If m₁m₂ = -1 then the two curves intersect orthogonally at P.
4. If m₁ = m₂ then the two curves touch each other at P.
Lengths of the tangent, normal, sub tangent and subnormal
1. Let the tangent at P to the curve meet x -a xis in T and the normal meet OX in N. Let PQ is drawn perpendicular to OX ;


then
(i) PT is called the length of the tangent at P.
(ii) PN is called the length of the normal at P.
(iii) QT, the projection of PT on OX is called sub tangent.
(iv) QN, the projection of PN on OX is called sub normal.
2. Let be a point on the curve y = f(x) and . Then
(i) Length of the tangent at P =
(ii) Length of the normal at P =
(iii) Length of the sub tangent at P =
(iv) Length of the sub normal at P =

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