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Let us take the critical value x= a. Find the sign of f '(x) for values of x slightly less than a and for values slightly greater than a. (i) If the sign of f '(x) changes from positive to negative as x increases through a, then f (a) is a local maximum value. (ii) If the sign of f '(x) changes fr..

f (x) = x 3 - 6x 2 + 9x + 15 f ' (x) = 3x 2 -12x + 9 = 3(x 2 - 4x + 3) = 3 (x - 1) (x - 3) Thus x = 1 and x = 3 are the only points which could be the points of local maxima or local minima. Let us examine for x=1 When x<1 (slightly less than 1) f '(x) = 3 (x - 1) (x - 3) = (+ ve) (- ve)..

For a particular Critical value x = a, find f " ' (a) (i) If f ''(a) < 0 then f (x) has a local maxima at x = a and f (a) is the maximum value. (ii) If f ''(a) > 0 then f (x) has a local minima at x = a and f (a) is the minimum value. (iii) If f ''(a) = 0 or , the test fails and the first d..

f '(x) = 6x 2 - 42x + 36 f '(x) = 0 x = 1 and x = 6 are the critical values f ''(x) =12x - 42 If x =1, f ''(1) =12 - 42 = - 30 < 0 x =1 is a point of local maxima of f (x). Maximum value = 2(1) 3 - 21(1) 2 + 36(1) - 20 = -3 If x = 6, f ''(6) = 72 - 42 = 30 > 0 x = 6 is a point..

Let f be a continuous function on an interval I = [a, b]. Then, f has the absolute maximum value and f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in ..

Find all the points where f ' takes the value zer..

At all the points calculate the values of ..

If a quantity y varies with respect to another quantity 'x' satisfying some rule y = f(x), in other words if y is a function x, then represents the rate of change of y with respect to x For x = x 0 , dy/dx at x 0 is called the rate of change of y with respect to x at x 0 . If y is a function of t x..

Let f be a real valued function in [a,b] such that f is continuous in [a,b]. f is differentiable in (a,b..