JEE Main & Advanced Mathematics Applications of Derivatives Higher Order Derivative Test

(1) Find \[f'(x)\]and equate it to zero. Solve \[f'(x)=0\]let its roots are \[x={{a}_{1}},{{a}_{2}}\].....

(2) Find  \[{f}''(x)\]and at \[x={{a}_{1}}\];

(i) If \[f''({{a}_{1}})\] is positive, then \[f(x)\] is minimum at \[x={{a}_{1}}\].

(ii) If \[f''({{a}_{1}})\] is negative, then \[f(x)\] is maximum at \[x={{a}_{1}}\].

(iii) If \[f''({{a}_{1}})=0\], go to step 3.

(3) If at \[x={{a}_{1}}\], \[f''({{a}_{1}})=0\], then find \[{f}'''(x)\]. If \[{f}'''({{a}_{1}})\ne 0\], then \[f(x)\]is neither maximum nor minimum at \[x={{a}_{1}}\].

If \[{f}'''({{a}_{1}})=0\], then find \[{{f}^{iv}}(x)\].

If \[{{f}^{iv}}(x)\] is \[+ve\] (Minimum value)

\[{{f}^{iv}}(x)\]is \[-ve\]  (Maximum value)

(4) If at \[x={{a}_{1}},\,\,{{f}^{iv}}({{a}_{1}})=0\], then find \[{{f}^{v}}(x)\] and proceed similarly.