Find the maximum and minimum values of the function f(x)=4 x+2 +x - Vidyadip

Question

Find the maximum and minimum values of the function $\text{f}(\text{x})=\frac{4}{\text{x}+2}+\text{x}$

✓

Answer

Given, $\text{f}(\text{x})=\frac{4}{\text{x}+2}+\text{x}$
$\Rightarrow\text{f}'(\text{x})=-\frac{4}{(\text{x}+2)^{2}}+1$
For a local maxima or a local minima, We must have f'(x) = 0
$\Rightarrow-\frac{4}{(\text{x}+2)^{2}}+1=0$
$\Rightarrow-\frac{4}{(\text{x}+2)^{2}}=-1$
$(\text{x}+2)^{2}=\pm2$
$\Rightarrow \text{x}=0 \ \text{and} -4$
Thus, x = 0 and x = -4 are the possible of local maxima or local minima.
Now, $\text{f}'(\text{x})=\frac{8}{(\text{x}+2)^{3}}$
At x = 0
$\text{f}''(0)=\frac{8}{(2)^{3}}=1>0$
So, x = 0 is a point of local minimum.
The local minimum value is given by
$\text{f}(0)=\frac{4}{(0+2)}+0=2$

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.(5 Marks)" from Maxima and Minima→Maxima and Minima — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\frac{\text{dx}}{(\text{x}^2+1)(\text{x}^2+4)}$

Solve the following differential equations:
$\text{x}\frac{\text{dy}}{\text{dx}}+\cot\text{y}=0,$ given that $\text{y}=\frac{\pi}{4},$ when $\text{x}=\sqrt{2}.$

A particle moves along the curve $y=x^2+2 x$. At what point(s) on the curve are the $x$ and $y$ coordinates of the particle changing at the same rate?

Solve the following differential equation:
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$

Solve the following system of equations by matrix method:
$x - y + 2z = 7$
$3x + 4y - 5z = -5$
$2x - y + 3z = 12$

Show that $\text{f(x)}=\begin{cases}12\text{x}-13, & \text{if x}\leq3\\2\text{x}^2+5, & \text{if x} > 3\end{cases}$ is differentiable at x = 3. Also, find f(3).

Differentiate $\tan^{-1}\Big(\frac{\sqrt{1+\text{x}^2}-1}{\text{x}}\Big)$ with respect to $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big),$ if $-1<\text{x}<1,\text{x}\neq0.$

If $\text{x}=2\cos\theta-\cot2\theta$ and $\text{y}=2\sin\theta-\sin2\theta,$ prove that $\frac{\text{dy}}{\text{dx}}=\tan\big(\frac{3\theta}{2}\big)$

Differentiate the following w.r.t. x:

$x^{e^x}+(\log x)^{\sin x}$

Evaluate the following intregals:
$\int\frac{1}{\text{x}[6(\log\text{x})^2+7\log\text{x}+2]}\ \text{dx}$