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

## Higher Order Derivative Test

Category : JEE Main & Advanced

(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.

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