Chapter 6

Application of Derivatives

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (37)

Examples

Exercise 6.1

Exercise 6.2

Exercise 6.2 · Q1Show that $3x+17$ is increasing on $\R$ worked
Exercise 6.2 · Q2Show that $e^{2x}$ is increasing on $\R$ worked
Exercise 6.2 · Q3 $\sin x$ rises on $\left(0,\frac\pi2\right)$ ⁠, falls on $\left(\frac\pi2,\pi\right)$ worked
Exercise 6.2 · Q4Where $2x^2-3x$ increases and decreasesworked
Exercise 6.2 · Q5Where $2x^3-3x^2-36x+7$ increases and decreasesworked
Exercise 6.2 · Q6Strictly increasing or decreasing: five functionsworked
Exercise 6.2 · Q7 $\log(1+x)-\frac{2x}{2+x}$ is increasing for $x>-1$ worked
Exercise 6.2 · Q8Where is $[x(x-2)]^2$ increasing?worked
Exercise 6.2 · Q9 $\frac{4\sin\theta}{2+\cos\theta}-\theta$ is increasing on $\left[0,\frac\pi2\right]$ worked
Exercise 6.2 · Q10 $\log x$ is increasing on $(0,\infty)$ worked
Exercise 6.2 · Q11 $x^2-x+1$ is neither increasing nor decreasing on $(-1,1)$ worked
Exercise 6.2 · Q12 (MCQ)Which of $\cos x$ ⁠, $\cos 2x$ ⁠, $\cos 3x$ ⁠, $\tan x$ decrease on $\left(0,\frac\pi2\right)$ ⁠?worked
Exercise 6.2 · Q13 (MCQ)On which interval is $x^{100}+\sin x-1$ decreasing?worked
Exercise 6.2 · Q14Values of $a$ making $x^2+ax+1$ increasing on $[1,2]$ worked
Exercise 6.2 · Q15 $x+\frac1x$ is increasing on any interval disjoint from $[-1,1]$ worked
Exercise 6.2 · Q16 $\log\sin x$ rises on $\left(0,\frac\pi2\right)$ and falls on $\left(\frac\pi2,\pi\right)$ worked
Exercise 6.2 · Q17 $\log|\cos x|$ falls on $\left(0,\frac\pi2\right)$ and rises on $\left(\frac{3\pi}2,2\pi\right)$ worked
Exercise 6.2 · Q18 $x^3-3x^2+3x-100$ is increasing on $\R$ worked
Exercise 6.2 · Q19 (MCQ)Where is $x^2e^{-x}$ increasing?worked

Exercise 6.3

Miscellaneous Exercise

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