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 +..
Theorem 3: (Second Derivative Test)
Let f be a differentiable function on an interval I and let a I. Let f "(a) be continuous at a. Then i) 'a' is a point of local maxima if f '(a) = 0 and f "(a) < 0 ii) 'a' is a point of local minima if f '(a) = 0 and f "(a) > 0 iii) The test fails if f '(a) = 0 and f "(a) = 0. In this case we..
Let f be a differentiable function on an interval I and let a I. Let f "(a) be continuous at a. Then i) 'a' is a point of local maxima if f '(a) = 0 and f "(a) < 0 ii) 'a' is a point of local minima if f '(a) = 0 and f "(a) > 0 iii) The test fails if f '(a) = 0 and f "(a) = 0. In this case we..Step 1
For a differentiable function f (x), find f '(x). Equate it to zero. Solve the equation f '(x) = 0 to get the Critical values of f (x..
Example:
Find the local maxima and local minima of the function f (x) = 2x 3 - 21x 2 + 36x - 20. Find also the local maximum and local minimum value..
Absolute Maximum and Absolute Minimum Value of a Function
Let f (x) be a real valued function with its domain D. (i) f(x) is said to have absolute maximum value at x = a if f(a) f(x) for all x D. (ii) f(x) is said to have absolute minimum value at x = a if f(a) f(x) for all x D. The following points are to be noted carefully with the help o..
Let f (x) be a real valued function with its domain D. (i) f(x) is said to have absolute maximum value at x = a if f(a) f(x) for all x D. (ii) f(x) is said to have absolute minimum value at x = a if f(a) f(x) for all x D. The following points are to be noted carefully with the help o..Theorem 5
Let f be a differentiable function on I and let x 0 be any interior point of I. Then (a) If f attains its absolute maximum value at x 0 , then f ' (x 0 )= 0 (b) If f attains its absolute minimum value at x 0 , then f '(x 0 ) = 0. In view of the above theorems, we state the following rule for find..
Seep 2:
Take the end points of the interva..
Step 4:
Take the maximum and minimum values of f out of the values calculated in step 3. These will be the absolute maximum or absolute minimum value..
Example:
A balloon which always remain spherical is being inflated by pumping in 900 cubic centimetres of gas per second. Final the rate at which the radius of the balloon increasing when the radius is 15cm. Let r and V be the variables for radius and volume of the balloon respectively. We have to find ..
A balloon which always remain spherical is being inflated by pumping in 900 cubic centimetres of gas per second. Final the rate at which the radius of the balloon increasing when the radius is 15cm. Let r and V be the variables for radius and volume of the balloon respectively. We have to find ..Geometrical meaning
Let A (a,f (a)) and B (b,f (b)) be two points on the graph of f (x) such that f(a) = f(b), then $ c (a, b) such that the tangent at P(c, f(c)) is parallel to x - axis..
Let A (a,f (a)) and B (b,f (b)) be two points on the graph of f (x) such that f(a) = f(b), then $ c (a, b) such that the tangent at P(c, f(c)) is parallel to x - axis..See what our Users say :
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