JEE Main & Advanced Mathematics Applications of Derivatives Definition

Definition

Category : JEE Main & Advanced

 

 

(1) A function \[f\] is said to be an increasing function in \[]\,a,\,b[,\] if \[{{x}_{1}}<{{x}_{2}}\Rightarrow f({{x}_{1}})<f({{x}_{2}})\] for all \[{{x}_{1}},{{x}_{2}}\in \,]\,a,b\,[.\]

 

 

(2) A function \[f\] is said to be a decreasing function in \[]\,a,\,b[,\] if \[{{x}_{1}}<{{x}_{2}}\Rightarrow f({{x}_{1}})>f({{x}_{2}})\], \[\,\,{{x}_{1}},{{x}_{2}}\in \,]\,a,\,b\,[.\]

 

 · \[f(x)\]is known as non-decreasing if \[f'(x)\ge 0\] and non-increasing if \[f'(x)\le 0\].

 

 

Monotonic function :  A function \[f\] is said to be monotonic in an interval if it is either increasing or decreasing in that interval.

 

 

We summarize the results in the table below :

 

               

\[{f}'(\,{{a}_{1}})\]

\[{f}''(\,{{a}_{1}})\]

\[{f}'''(\,{{a}_{1}})\]

Behaviour of \[f\] at \[{{a}_{1}}\]

+

 

 

Increasing

\[-\]

 

 

Decreasing

0

+

 

Minimum

0

\[-\]

 

Maximum

0

0

 

-

0

0

\[\mp \]

Inflection

 

0

0

-

 

 

 · Blank space indicates that the function may have any value at a1.

 

· Question mark indicates that the behaviour of \[f\] cannot be inferred from the data.

 

 

Properties of monotonic functions

 

 

(i) If \[f(x)\]is a strictly increasing function on an interval \[[a,\,\,b]\], then \[\psi =\frac{\pi }{2}\Rightarrow {{\left( \frac{dy}{dx} \right)}_{({{x}_{1}},\,{{y}_{1}})}}\to \,\,\,\infty \] exists and it is also a strictly increasing function.

 

 

(ii) If \[f(x)\] is a strictly increasing function on an interval \[[a,\,\,b]\] such that it is continuous, then \[{{f}^{-1}}\] is continuous on \[[f(a),f(b)]\].

 

 

(iii) If \[f(x)\] is continuous on \[[a,\,\,b]\] such that \[f'(c)\ge 0\] \[[f'(c)>0]\] for each \[c\in (a,\,\,b)\], then \[f(x)\] is monotonically (strictly) increasing function on \[[a,\,\,b]\].

 

 

(iv) If\[f(x)\] is continuous on \[[a,\,\,b]\] such that \[f'(c)\le 0\] \[[f'(c)>0]\] for each \[c\in (a,\,b)\], then \[f(x)\] is monotonically (strictly) decreasing function on \[[a,\,\,b]\].

 

 

(v) If \[f(x)\] and g(x) are monotonically (or strictly) increasing (or decreasing) functions on \[[a,\,\,b]\], then \[gof(x)\] is a monotonically (or strictly) increasing function on \[[a,\,\,b]\].

 

 

(vi) If one of the two functions \[f(x),\,\,g(x)\] is strictly (or monotonically) increasing and other a strictly (monotonically) decreasing, then \[gof(x)\] is strictly (monotonically) decreasing on \[[a,\,\,b]\].

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