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Step 1:
Show the function f (x) is continuous on the closed interval [a, b..
Step 3:
If the above condition are satisfied, then Mean Value Theorem is applicabl..
Example:
Verify mean value theorem for the function f (x) = (x - 4) (x - 6) (x - 8) in [4,..
Step 2:
f ' (x) = (x - 6) (x - 4) + (x - 4) (x - 8) + (x - 6) (x - 8) f '(x)= (x 2 -10x + 24) + (x 2 - 12x + 32)+ (x 2 - 14x + 48) = 3x 2 - 36x + 104 f '(x) is defined for all values on the interval (4,10). \ f '(x) is differentiabl..
Maxima and Minima
A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that f(a) f(x) for all x I The number f(a) is called the local maximum of f(x). The point a is called the point of maximum. Note that when 'a' is the point of local maxima, f(x) is increasing for all va..
A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that f(a) f(x) for all x I The number f(a) is called the local maximum of f(x). The point a is called the point of maximum. Note that when 'a' is the point of local maxima, f(x) is increasing for all va..Theorem 2:
(First Derivative Test) Let f (x) be a real valued differentiable function. Let a be a point on an interval I such that f '(a) = 0. (a) a is a local maxima of the function f (x) if i) f (a) = 0 ii) f (x) changes sign from positive to negative as x increases through a. That is, f (x) > 0 for x &l..
(First Derivative Test) Let f (x) be a real valued differentiable function. Let a be a point on an interval I such that f '(a) = 0. (a) a is a local maxima of the function f (x) if i) f (a) = 0 ii) f (x) changes sign from positive to negative as x increases through a. That is, f (x) > 0 for x &l..Step 1:
Find f '(..
Step 3:
Check the sign of f'(x) in the immediate neighbourhood of each critical valu..
Example:
Find the local maxima or local minima, if any, for the following function using first derivative test f (x) = x 3 - 6x 2 + 9x +..
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