solving calculus problems step by step

Solve f '(x) = 0 to get the critical values for f (x). Let these values be a, b, c. These are the points of maxima or minima. Arrange these values in ascending ord..

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..

Express the differential equation in the form f(x) dx = g(y) d..

Show the function f (x) is continuous on the closed interval [a, b..

If the above condition are satisfied, then Mean Value Theorem is applicabl..

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..

Find f '(..

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..

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

Show that the function is continuous in the given interval. Some known standard functions which are continuous, can be mentioned directl..