**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 f^{iv }(a).

If f^{iv}(a)
= positive, then f(x) is minimum at x = a.

If f^{iv}(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).