Let \[f(x)\] be a function differentiable at \[x=a\].
Then (a) \[x=a\]is a point of local maximum of \[f(x)\] if
(i) \[f'(a)=0\] and
(ii) \[f'(a)\]changes sign from positive to negative as \[x\] passes through \[a\] i.e., \[f'(x)>0\] at every point in the left neighbourhood \[(a-\delta ,a)\] of \[a\] and \[f'(x)<0\] at every point in the right neighbourhood \[(a,\,\,a+\delta )\] of \[a\].
(b) \[x=a\] is a point of local minimum of \[f(x)\] if
(i) \[f'(a)=0\]and
(ii) \[f'(a)\] changes sign from negative to positive as \[x\] passes through \[a,\] i.e., \[f'(x)<0\] at every point in the left neighbourhood \[(a-\delta ,a)\] of \[a\] and \[{{A}_{1}}=\frac{1}{3}(2a+b),\,{{A}_{2}}=\frac{1}{3}(a+2b)\] at every point in the right neighbourhood \[(a,a+\delta )\]of \[a\].
(c) If \[f'(a)=0\] but \[f'(a)\] does not change sign, that is, has the same sign in the complete neighbourhood of \[a,\] then \[a\] is neither a point of local maximum nor a point of local minimum.
Working rule for determining extreme values of a function \[f(x)\]
Step I : Put \[y=f(x)\]
Step II : Find \[\frac{dy}{dx}\]
Step III : Put \[\frac{dy}{dx}=0\] and solve this equation for \[x\]. Let \[x={{c}_{1}},{{c}_{2}},.....,{{c}_{n}}\] be values of \[x\] obtained by putting \[\frac{dy}{dx}=0.\] \[{{c}_{1}},\,{{c}_{2}},\,.........{{c}_{n}}\] are the stationary values of \[x\].
Step IV : Consider \[x={{c}_{1}}\].
If \[\frac{dy}{dx}\] changes its sign from positive to negative as \[x\] passes through \[{{c}_{1}}\], then the function attains a local maximum at \[x={{c}_{1}}\]. If \[\frac{dy}{dx}\] changes its sign from negative to positive as \[x\] passes through \[{{c}_{1}}\], then the function attains a local minimum at \[x={{c}_{1}}\]. In case there is no change of sign, then \[x={{c}_{1}}\] is neither a point of local maximum nor a point of local minimum.