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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