Test for Local Maximum/ Minimum – Second Derivative Test
Second Derivative Test: First we find the roots of f’(x) = 0. Suppose x = a is one of the roots of f’(x) = 0
Now, find f’’(x) at x = a
(a)If f’’(a) = negative, then f(x) is maximum at x = a.
(b) If f’’(x) = positive, then f(x) is minimum at x = a.
(c) If f’’(x) = zero, then we find f’’’(x) at x = a.
If f’’’(x) ≠ 0, then f(x) has neither maximum nor minimum (inflexion point) at x = a
But, if f’’’(x) ≠ 0, then find fiv (a).
If fiv(a) = positive, then f(x) is minimum at x = a.
If fiv(a) = negative, then f(x) is maximum at x = a and so on
Example: Locate the position and nature of any turning point of the function y = x³ – 3x + 2 is
Solution: We need find where the turning point are and whether maximum and minimum points. First of all, we carry out the differentiation and set d y /dx equal to zero.
y = x³ – 3x + 2
dy/dx = 3x² – 3(1) + 0
= 3x² – 3
=3(x² – 1) … (1)
At stationary points dy/dx = 0
3(x² – 1) = 0
x² – 1 = 0
x² = 1
x = 1, -1
y = x³ – 3x + 2
put x = 1
y = (1)³ – 3(1) + 2
= 0
(x, y) = (1, 0)
y = x³ – 3x + 2
put x = -1
= (-1)³ – 3(-1) + 2
= 4
(x, y) = (-1, 4)
Equation (1) differentiation with respect x
d²y/ dx² = 6x – 0
= 6x
Put x = 1
6(1) = 6
Put x = -1
6(-1) = -6
Hence minima at (1, 0).