Arithmetic-Geometric-Harmonic Mean Inequality

The Arithmetic-Geometric-Harmonic Mean of positive real numbers is defined as follows

Arithmetic Mean \(\ge\) Geometric Mean \(\ge\) Harmonic Mean

(i) If \(a,\ b\ \gt\ 0\ \ then\ \ \frac{a+b}{2}\ \ge\ \sqrt{ab}\ \ge\ \frac{2}{\frac{1}{a}+\frac{1}{b}}\)

(ii) If \(a_i\ \gt\ 0,\) where \(i\ =\ 1,\ 2, \ 3, ..., n,\) then

     \(\frac{a_1\ +\ a_2\ +\ ...+\ a_n}{n}\ \ge\ (a_1\ \cdot\ a_2\ ...\ a_n)^{1/n}\ \ge\ \frac{n}{\frac{1}{a_1}\ +\ \frac{1}{a_2}\ +\ ...+\ \frac{1}{a_n}}\)

Logarithm Inequality

If \(a \) is a positive real number other than \(1\ \ and\ \ a^x=m,\) then \(x\) is called the logarithm of \(m\) to the base \(a\), written as \(log_a\ m\). In \(log_a\ m,\ m\) should always be positive.

   (i) If \(m\le0,\) then \(log_a\ m\) will be meaningless.

   (ii) \(log_a\ m\) exists, if \(m,\ a\gt 0\) and \(a \ne 1\).

Important Results on Logarithm

   \((i)\ \ a^{{\log_a}x}=x;\ a\gt 0,\ \ne 1,\ x \gt 0\)

   \((ii)\ \ a^{{\log_b}x}= x^{{\log_b}a};\ a,\ b \gt 0,\ \ne1,\ x \gt 0\)

   \((iii)\ \ a^{{\log_a}a}=1,\ a \gt0,\ \ne 1\)

   \((iv)\ \ \log_a\ x=\frac{1}{log_x\ a};\ x,\ a \gt 0,\ \ne 1\)

   \((v)\ \ \log_a\ x= \log_a\ b\ \ \log_b\ x= \frac{\log_b\ x}{\log_b\ a};\ a,\ b \gt 0,\ \ne 1,\ x \gt 0\)

    \((vi)\ \ \text{For}\ x \gt 0;\ a \gt 0,\ \ne 1\\ \ \ \ \ \ \ \ (a)\ \log_{a^n}(x)= \frac{1}{n}\ \log_a\ x\\ \ \ \ \ \ \ \ (b)\ \log_{a^n}\ x^m=\big(\frac{m}{n}\big)\ \log_a\ x\)

\((vii)\ \ \text{For}\ x \gt y \gt 0\\ \ \ \ \ \ \ \ \ (a)\ \log_a\ x \gt \log_a\ y,\ \text{if}\ a \gt 1\\ \ \ \ \ \ \ \ \ (b)\ \log_a\ x \lt \log_a\ y,\ \text{if}\ 0\ \lt a \lt 1\)

\((viii)\ \ \text{If}\ a \gt 1\ \text{and}\ x \gt 0,\ \text{then}\\ \ \ \ \ \ \ \ \ (a)\ \log_a\ x \gt p\ \Rightarrow x \gt a^p\\ \ \ \ \ \ \ \ \ (b)\ 0 \lt \log_a\ x \lt p\ \Rightarrow 0 \lt x \lt a^p\)

\((ix)\ \ \text{If}\ \ 0 \lt a \lt 1,\ \text{then}\\ \ \ \ \ \ \ \ \ (a)\ \log_a\ x \gt p\ \Rightarrow 0 \lt x \lt a^p\\ \ \ \ \ \ \ \ \ (b)\ 0 \lt \log_a\ x \lt p \Rightarrow 0\)

Quadratic Equation

\(\bullet\)  An equation of the form \(ax^2\ +\ bx\ +\ c=0,\) where \(a \ne0,a, b\ \ and\ \ c, x \in R,\) is called a real quadratic equation. Here \(a, b, \ \ and\ \ c\) are called the coefficients of the equation.

\(\bullet\)  The quantity \(D=\ b^2\ -\ 4\ ac\) is known as the discriminant of the equation \( \ ax^2\ +\ bx\ +\ c=0\) and its roots are given by \(x=\frac{-b\ \pm\ \sqrt D}{2a}\)

\(\bullet\)  An equation of the form \(az^2\ +\ bz\ +\ c=0,\) where \(a \ne0,a, b\ \ and\ \ c, z \in C\ (\text{complex}),\) is called a complex quadratic equation and its roots are given by \(z=\frac{-b\ \pm\ \sqrt D}{2a}\)

Nature of Roots of Quadratic Equation

Let \(a, b, c\ \in\ R\ \ and\ \ a \ne0,\) then the equation \(ax^2\ +\ bx\ +\ c=0\)

(i) has real and distinct roots if and only if \(D \gt 0.\)

(ii) has real and equal roots if and only if \(D=0\).

(iii) has complex roots with non-zero imaginary parts if and only if

Some Important Results

(i)  If \(p\ +\ iq\ \ (\text{where} \ p,q\ \in\ R,\ q\ \ne0)\) is one root of \(ax^2\ +\ bx\ +\ c\ =0\), then second root will be \(p\ -\ iq\).

(ii)  If \(a,\ b,\ c \in Q\ \ and\ \ p\ +\ \sqrt q\) is an irritational root of \(ax^2\ +\ bx\ +\ c=0\), then other root will be \(p\ -\ \sqrt q\).

(iii)  If \(a,\ b,\ c \in Q\ \ and\ \ D\) is a perfect square, then \(ax^2\ +\ bx\ +\ c\ =0,\) has rational roots.

(iv)  If \(a=1,\ b,\ c \in\ I\) and roots of \(ax^2\ +\ bx\ +\ c\ =0\) are rational numbers, then these roots must be integers.

(v)  If the roots of \(ax^2\ +\ bx\ +\ c\ =0\) are both positive, then the signs of \(a\ \ and\ \ c\) should be like and opposite to the sign of \(b\).

(vi)  If the roots of \(ax^2\ +\ bx\ +\ c\ =0\) are both negative, then signs of \(a,\ b\ \ and\ \ c\) should be like.

(vii)  If the roots of \(ax^2\ +\ bx\ +\ c\ =0\) are reciprocal to each other, then \(c=a.\)

(viii)  In the equation \(ax^2\ +\ bx\ +\ c\ =0\ (a, b, c \in R).\)

        If \(a\ +\ b\ +\ c=0\), then the roots are \(1,\ \frac{c}{a}\) and if \(a\ -\ b\ +\ c=0,\) then the roots are \(-1\ \ and\ \ -\frac{c}{a}.\)

Relation between Roots and Coefficients

Quadratic Roots

If \(\alpha \ \ and\ \ \beta \) are the roots of quadratic equation \(ax^2\ +\ bx\ +\ c=0:\ a\ \ne\ 0,\) then sum of roots \(=\ \alpha\ +\ \beta=\ -\frac{b}{a}\)

and product of roots \(=\alpha\ \beta=\frac{c}{a}.\)

And, also \(ax^2\ +\ bx\ +\ c\ =0\ (x-\alpha)\ (x-\beta)\)

Cubic Roots

If \(\alpha,\ \beta\ \ and\ \ \gamma\) are the roots of cubic equation \(ax^3\ +\ bx^2\ +\ cx\ +\ d=0;\ a\ \ne\ 0,\) then \(\alpha\ +\ \beta\ +\ \gamma=\ -\frac{b}{a}\)

            \(\beta\ \gamma\ +\ \gamma\ \alpha\ +\ \alpha\ \beta=\ \frac{c}{a}\)

and      \(\alpha\ \beta\ \gamma=\ -\frac{d}{a} \)

Common Roots (Conditions)

Suppose that the quadratic equations are \(ax^2\ +\ bx\ +\ c=0\ \ and\ \ a'\ x^2\ +\ b'\ x\ +\ c=0.\)

   (i)  When one root is common, then the condition is 

         \((a'c\ -\ a\ c')^2\ =\ (b\ c'\ -\ b'\ c)\ (a\ b'\ -\ a'\ b).\)

   (ii)  When both roots are common, then the condition is 

          \(\frac{a}{a'}\ =\ \frac{b}{b'}\ =\frac{c}{c'}\)

Formation of an Equation

\(Quadratic\ Equation\)

If the roots of a quadratic equation are \(\alpha\ \ and\ \ \beta\), then the equation will be of the form \(x^2\ -\ (\alpha\ +\ \beta)\ x\ +\ \alpha\ \beta=0\).

\(Cubic\ Equation\)

If \(\alpha,\ \beta\ \ and\ \ \gamma\) are the roots of the cubic equation, then the equation will be form of   \(x'-\ (\alpha\ +\ \beta\ +\ \gamma)\ x^2\ +\ (\alpha\ \beta\ +\ \beta\ \gamma\ +\ \gamma\ \alpha)\ x\ -\ \alpha\ \beta\ \gamma=0\).

Transformation of Equations

Let the given equation be \(a_0 x^n\ +\ a_1 x^{n-1}\ +\ ...\ +\ a_{n-1}\ x\ +\ a_n=0\ \ ..... \ (A)\)

Then the equation

   (i)  whose roots are \(k\ (\ne0)\) times roots of the Eq. (A), is obtained by replacing \(x\) by \(\frac{x}{k}\) in Eq. (A).

   (ii)  whose roots are the negatives of the roots of Eq. (A), is obtained by replacing \(x\) by \(-x\) in Eq. (A).

   (iii)  whose roots are \(k\) more than the roots of Eq. A, is obtained by replacing \(x\) by \((x-k)\) in Eq. (A).

   (iv)  whose roots are reciprocals of the roots of Eq. (A), is obtained by replacing \(x\) by \(1/x\) in Eq. (A) and then multiply both the sides by \(x^n.\)

Maxima and Minimum Value \(ax^2\ +\ bx\ +\ c\)

   (i)  When \(a\ \gt\ 0,\) then minimum value of \(ax^2\ +\ bx\ +\ c\) is 

            \(\frac{-D}{4a}\ \ or\ \ \frac{4ac-b^2}{4a}\) at  \(x=\frac{-b}{2a}\)

  (ii)  When \(a\ \lt\ 0,\) then maximum value of \(ax^2\ +\ bx\ +\ c\)

             \(\frac{-D}{4a}\ \ or\ \ \frac{4ac-b^2}{4a}\) at \(x=\frac{-b}{2a}\)

Sign of Quadratic Expression

Let \(f(x)=ax^2\ +\ bx\ +\ c,\) where \( a,\ b\ and\ \ c\in R \ \ and\ \ a \ne 0.\)

   (i)   If \(a \gt 0\ \ and\ \ D \lt 0,\ \ then\ f(x) \gt0,\ \forall x \in R \).

   (ii)   If \(a \lt 0\ \ and\ \ D \lt 0,\ \ then\ f(x) \lt0,\ \forall x \in R \).

   (iii)   If \(a \gt 0\ \ and\ \ D = 0,\ \ then\ f(x) \ge0,\ \forall x \in R \).

   (iv)   If \(a \lt 0\ \ and\ \ D = 0,\ \ then\ f(x) \le0,\ \forall x \in R \).

   (v)   If \(a\ \gt\ 0,\ D \ \gt\ 0\ \ and\ f(x)=0\) have two real roots \(\alpha\ \ and\ \ \beta\), where \((\alpha \lt \beta),\) then \( f(x)\ \gt0,\ \forall\ x\ \in\ (- \infty, \alpha)\ \cup\ (\beta, \infty),\ and\ f(x)\ \lt 0,\ \forall\ x\ \in\ (- \alpha, \beta).\)

  (vi)  If \(a\ \lt\ 0,\ D \ \gt\ 0\ \ and\ f(x)=0\) have two real roots \(\alpha\ \ and\ \ \beta,\) then \( f(x)\ \lt 0,\ \forall\ x\ \in\ (- \infty, \alpha)\ \cup\ (\beta, \infty),\ and\ f(x)\ \gt 0,\ \forall\ x\ \in\ ( \alpha, \beta).\)

Position of Roots

Let \(ax^2\ +\ bx\ +\ c=0\) has roots \(\alpha\ \ and\ \ \beta\). Then. we have the following conditions:

   (i)  with respect to one real number (k).

(ii)  with respect to two real numbers \(k_1\ \ and\ \ k_2\)

(iv) If \(a\ \lt\ 0\ \lt\ b\), then

      \((a)\ \ a^2\ \gt\ b^2,\ \text{if}\ \ |a|\ \gt\ |b|\ \ \ \ \ \ (b)\ \ a^2\ \lt\ b^2,\ \text{if}\ \ |a|\ \lt\ |b|\)

(v) If \(a\ \lt\ x\ \lt\ b\) and \(a,\ b\) are positive real numbers, then 

     \(a^2\ \lt\ x^2\ \lt\ b^2.\)\(\)

(vi) If \(a\ \lt\ x\ \lt\ b\) and \(a\) is negative number and \(b\) is positive number,  then

    \((a)\ \ 0\ \le\ x^2\ \lt\ b^2,\ \text{if}\ \ |b|\ \gt\ |a|\ \ \ \ \ \ (b)\ \ 0\ \le\ x^2\ \lt\ a^2,\ \text{if}\ \ |a|\ \gt\ |b|\)

(vii) If \(a_i\ \gt\ b_i\ \gt\ 0,\) where \(i=1,\ 2,\ 3,\ ...,n,\) then

       \(a_1\ a_2\ a_3\ ... a_n\ \gt\ b_1\ b_2\ b_3\ ... b_n.\)

(viii)  If \(a_i\ \gt\ b_i,\) where \(i=1,\ 2,\ 3,\ ...,n,\)then

         \(a_1\ +\ a_2\ +\ a_3\ +... +\ a_n\ \gt\ b_1\ +\ b_2\ +\ ... +\ b_n.\)

(ix) If \(|x|\ \lt\ a,\) then

      (a) for \(a\ \gt\ 0,\ -a\ \lt\ x\ \lt\ a.\)

      (b) for \(a\ \lt\ 0,\ x\ \in\ \phi\).

Inequalities

Let \(a\ \ and \ \ b\) be two real numbers. if \(a-b\) is negative, we say that \(a\) is less than \(b\ \ (a\ \lt\ b)\) and if \(a-b\) is positive. then \(a \) is greater than \(b\ \ (a\ \gt\ b)\). This shows the inequalities concept.

Important Results on Inequalities

    (i) If \(a\ \gt\ b\), then \(a\ \pm\ c\ \gt\ b\ \pm\ c,\ \forall\ c\ \in\ R\).

    (ii) If \(a\ \gt\ b\), then

         (a) for \(m\ \gt\ 0,\ am\ \gt\ bm,\ \frac{a}{m}\ \gt\ \frac{b}{m}\)

         (b) for \(m\ \lt\ 0,\ am\ \lt\ bm,\ \frac{a}{m}\ \lt\ \frac{b}{m}\)

    (iii)  (a) If \(a\ \gt\ b\ \gt\ c,\) then

                \(*\ a^2\ \gt\ b^2\ \ \ *\ |a|\ \gt\ |b|\ \ \ *\ \frac{1}{a}\ \lt\ \frac{1}{b}\)

          (b) If \(a\ \lt\ b\ \lt\ c,\) then

                \(*\ a^2\ \gt\ b^2\ \ \ *\ |a|\ \gt\ |b|\ \ \ *\ \frac{1}{a}\ \gt\ \frac{1}{b}\)