First Derivative Test for Maxima and Minima

Here you will learn first derivative test for maxima and minima with examples.

Let’s begin –

First Derivative Test for Maxima and Minima

If f'(x) = 0 at a point (say x = a) and 

(i) If f'(x) changes sign from positive to negative in the neighbourhood of x = a then x = a is said to be a point local maxima.

(ii) If f'(x) changes sign from negative to positive in the neighbourhood of x = a then x = a is said to be a point local minima.

Note – If f'(x)  does not change sign i.e. has the same sign in a certain complete neighbourhood of a, then f(x) is either increasing or decreasing throughout this neighbourhood implying that x = a is not a point of extremum of f.

Example : Let f(x) = x + \(1\over x\) ; x \(\ne\) 0. Discuss the local maximum and local minimum values of f(x)

Solution : Here f'(x) = 1 – \(1\over x^2\) = \(x^2 – 1\over x^2\)

= \((x – 1)(x + 1)\over x^2\)

Using local number line rule, line

f(x)  will have local maximum at x = – 1 and local minimum at x =1 

\(\therefore\) local maximum value of f(x) = -2 at x = -1

and local minimum value of f(x) = 2 at x = 1

Example : Let f(x) = \(x^3 – 3x\). Discuss the local maximum and local minimum values of f(x)

Solution : Here f'(x) = \(3x^2 -3\) = 3(x + 1)(x – 1)

Using local number line rule, line

f(x)  will have local maximum at x = – 1 and local minimum at x =1 

\(\therefore\) local maximum value of f(x) = 2 at x = -1

and local minimum value of f(x) = -2 at x = 1

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