Show that the function f(x)=x-2 x+1 , x -1 is increasing. - Vidyadip

Question

Show that the function $f(x)=\frac{x-2}{x+1}, x \neq-1$ is increasing.

✓

Answer

$
\begin{aligned}
& f(x)=\frac{x-2}{x+1} \\
& \therefore f^{\prime}(x)=\frac{d}{d x}\left(\frac{x-2}{x+1}\right)=\frac{(x+1) \cdot \frac{d}{d x}(x-2)-(x-2) \cdot \frac{d}{d x}(x+1)}{(x+1)^2}
\end{aligned}
$
$
\begin{gathered}
\quad=\frac{(x+1) \cdot(1-0)-(x-2) \cdot(1+0)}{(x+1)^2} \\
=\frac{x+1-x+2}{(x+1)^2}=\frac{3}{(x+1)^2} \\
\because x \neq-1 \quad \therefore x+1 \neq 0 \\
\therefore(x+1)^2>0 \quad \therefore \frac{3}{(x+1)^2}>0
\end{gathered}
$
$\therefore f ^{\prime}( x )>0$, for all $x \in R , x \neq-1$
Hence, the function $f$ is increasing for all $x \in R$, where $x \neq-1$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(2 Marks)" from Applications of Derivatives (p-1)→Applications of Derivatives (p-1) — all question types→Maths (commerce) STD 12 Commerce / Arts question papers→Maharashtra Board STD 12 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

Find the adjoint of the following matrices : $\left[\begin{array}{cc}2 & -3 \\ 3 & 5\end{array}\right]$

Rewrite the following statements without using conditional:
[Hint: P → q ≡ ~p ∨ q]
(i) If price increases, then demand falls.
(ii) If demand falls, then the price does not increase.

Determine whether the following statement patterns is a tautology or a contradiction or a contingency:
[(~p ∧ q) ∧ (q ∧ r)] ∨ (~q)

Find the accumulated value of annuity due of $\text{₹} 1,000$ p.a. for $3$ years at $10 \%$ p.a. compounded annually. [Given: $(1.1)^3=1.331$ ]

If Laspeyre’s Price Index Number is four times Paasche’s Price Index Number, then find the relation between Dorbish-Bowley’s and Fisher’s Price Index Numbers.

Evaluate the following definite integrals : $\int_0^1 \frac{x^2+3 x+2}{\sqrt{x}} d x$

Examine whether the statement pattern

[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.

A doctor has prescribed two different kinds of feeds A and B to form a weekly diet for a sick person. The minimum requirement of fats, carbohydrates, and proteins are 18, 28,14 units respectively. One unit of food A has 4 units of fat, 14 units of carbohydrates, and 8 units of protein. One unit of food B has 6 units of fat, 12 units of carbohydrates and 8 units of protein. The price of food A is ₹ 4.5 per unit and that of food B is ₹ 3.5 per unit. Form the LPP so that the sick person’s diet meets the requirements at minimum cost.

If $A=\left[\begin{array}{cc}1 & -2 \\ 5 & 3\end{array}\right], B=\left[\begin{array}{cc}1 & -3 \\ 4 & -7\end{array}\right]$, then find the matrix $A-2 B+6 I$, where $I$ is the unit matrix of order 2 .

Find the values of $x$, such that $f(x)$ is increasing function:
$f(x)=x^2+2 x-5$