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Application of Derivatives, , , For a quantity y varying with another quantity x, satisfying the rule y = f(x), the rate of, change of y with respect to x is given by, The rate of change of y with respect to x at the point x = x0 is given by, , , , If the variables x and y are expressed in form of x = f(t) and y = g(t), then the rate of change, of y with respect to x is given by, , a. A function f: (a, b) → R is said to be, , , , increasing on (a, b), if x1 < x2 in (a, b), decreasing on (a, b), if x1 < x2 in (a, b), OR, If a function f is continuous on [a, b] and differentiable on (a, b), then, , , , f is increasing in [a, b], if, , for each x (a, b), , , , f is decreasing in[a, b], if, , for each x (a, b), , , , f is constant function in [a, b], if, , a. A function f: (a, b), , , , ., , for each x (a, b), , R is said to be, , strictly increasing on (a, b), if x1 < x2 in (a, b) f(x1) < f(x2) x1, x2 (a, b), strictly decreasing on (a, b), if x1 < x2 in (a, b) f(x1) > f(x2) x1, x2 (a, b), , c. The graphs of various types of functions can be shown as follows:

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Example 1: Find the intervals in which the function f given, by, , is strictly increasing or decreasing., , Solution:, , The points, , and, , divide the interval [0, 2π] into three disjoint, , intervals,, , ., , Now,, f is strictly increasing in the intervals, Also,, , .

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f is strictly decreasing in the interval, , , For the curve y = f(x), the slope of tangent at the point (x0, y0) is given, by, , , , ., , ., , For the curve y = f(x), the slope of normal at the point (x0, y0) is given, by, , ., , , , The equation of tangent to the curve y = f(x) at the point (x0, y0) is given, by,, , , , If, does not exist, then the tangent to the curve y = f(x) at the point (x0, y0) is parallel, to the y-axis and its equation is given by x = x0., , , , The equation of normal to the curve y = f(x) at the point (x0, y0) is given, by,, , , , If, does not exist, then the normal to the curve y = f(x) at the point (x0, y0) is parallel, to the x-axis and its equation is given by y = y0., , , , If, = 0, then the respective equations of the tangent and normal to the curve y = f(x) at, the point (x0, y0) are y = y0 and x = x0., , , , Let y = f(x) and let Δx be a small increment in x and Δy be the increment in y corresponding, to the increment in x i.e., Δy = f(x + Δx) – f(x), Then,, is a good approximation of Δy, when dx = Δx is, relatively small and we denote it by dy ≈ Δy., , o, , o, o, , Maxima and Minima: Let a function f be defined on an interval I. Then, f is said to have, maximum value in I, if there exists c ∈ I such that f(c) > f(x), ∀ x ∈ I [In this case, c is called, the point of maxima], minimum value in I, if there exists c ∈ I such that f(c) < f(x), ∀ x ∈ I [In this case, c is called, the point of minima], an extreme value in I, if there exists c ∈ I such that c is either point of maxima or point of, minima [In this case, c is called an extreme point], Note: Every continuous function on a closed interval has a maximum and a minimum value.

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, , Local maxima and local minima: Let f be a real-valued function and c be an interior point, in the domain of f. Then, c is called a point of, , o, , local maxima, if there exists h > 0 such that f(c) > f(x), ∀ x ∈ (c – h, c + h) [In this case, f(c) is, called the local maximum value of f], local minima, if there exists h > 0 such that f(c) < f(x), ∀ x ∈ (c – h, c + h) [In this case, f(c) is, called the local maximum value of f], , o, , , , A point c in the domain of a function f at which either, called a critical point of f., , , , First derivative test: Let f be a function defined on an open interval I. Let f be continuous, at a critical point c in I. Then:, , o, , If, changes sign from positive to negative as x increases through c, i.e. if, at, every point sufficiently close to and to the left of c, and, at every point sufficiently, close to and to the right of c, then c is a point of local maxima., , o, , If, changes sign from negative to positive as x increases through c, i.e. if, at, every point sufficiently close to and to the left of c, and, at every point sufficiently, close to and to the right of c, then c is a point of local minima., , o, , If, does not change sign as x increases through c, then c is neither a point of local, maxima nor a point of local minima. Such a point c is called point of inflection., , , , Second derivative test: Let f be a function defined on an open interval I and c ∈ I. Let f be, twice differentiable at c and, Then:, , o, , If, of f., If, of f., , o, , or f is not differentiable is, , , then c is a point of local maxima. In this situation, f(c) is local maximum value, , then c is a point of local minima. In this situation, f(c) is local minimum value

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o, , If, , then the test fails. In this situation, we follow first derivative test and find, whether c is a point of maxima or minima or a point of inflection., Example 1: Find all the points of local maxima or local minima of the function f given, by f(x) = x3 – 12x2 + 36x – 4., Solution:, We have,, , Therefore, the point of local maxima and local minima are at the points x = 2 and x = 6, respectively., The local maximum value is f(2) = 28, The local minimum value is f(6) = –4