JEE Main & Advanced Mathematics Applications of Derivatives Definition

(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 :

 · 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 \[[a,\,\,b]\].