Theorem 4
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 ..
Step 1:
Find all the points where f ' takes the value zer..
Step 3:
At all the points calculate the values of ..
Rate of Change of Quantity
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..
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..Rolle's Theorem
Let f be a real valued function in [a,b] such that f is continuous in [a,b]. f is differentiable in (a,b..
Let f be a real valued function in [a,b] such that f is continuous in [a,b]. f is differentiable in (a,b..Note 1:
We cannot obtain c if any one of the conditions of Rolle's theorem are not satisfie..
Example:
Verify Rolle's theorem for the function f (x) = x 2 - 8x + 12 on (2, 6) Since a polynomial function is continuous and differentiable everywhere f (x) is differentiable and continuous (i) and (ii) conditions of Rolle's theorem is satisfied. f (2) = 2 2 - 8 (2) + 12 = 0 f (6) = 36 - 48 + 12 = 0 ..
Verify Rolle's theorem for the function f (x) = x 2 - 8x + 12 on (2, 6) Since a polynomial function is continuous and differentiable everywhere f (x) is differentiable and continuous (i) and (ii) conditions of Rolle's theorem is satisfied. f (2) = 2 2 - 8 (2) + 12 = 0 f (6) = 36 - 48 + 12 = 0 ..Step 1:
Show that the function is continuous in the given interval. Some known standard functions which are continuous, can be mentioned directl..
Step 3:
Check if f (a) = f (b) If all the above condition are satisfied, then Rolle's theorem is applicable else the Rolle's theorem is not applicable. If Rolle's theorem is applicable, solve f '(c) = 0. Show that one of these roots lie in the open interval (a, b..
Note 1:
If m = 0 the tangent is parallel to x-axis. ..
If m = 0 the tangent is parallel to x-axis. ..See what our Users say :
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