Write the minimum value of f(x)=x+1 x ,x>0. - Vidyadip

Question

Write the minimum value of $\text{f}(\text{x})=\text{x}+\frac{1}{\text{x}},\text{x}>0.$

✓

Answer

Given, $\text{f}\text{(x)}=\text{x}+\frac{1}{\text{x}}$
Implies that $\text{f}'\text{(x)}=1-\frac{1}{\text{x}^2}$
For a local maxima or a local minima, We must have
$\text{f}'\text{(x)}=0$
Implies that $1-\frac{1}{\text{x}^2}=0$
Implies that $\text{x}^2=1$
Implies that $\text{x}=1,-1$
But $\text{x}>0$
Implies that $\text{x}=1$
Now, $\text{f}''(\text{x})=\frac{1}{\text{x}^{3}}$
At x = 1
$\text{f}''(1)=\frac{2}{(1)^{3}}1=2>0$
Therefore, x = 1 is a point minimum.
Thus, the local minimum value is given by
$\text{f}(1)=1+\frac{1}{1}=1+1=2$

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 "3 Marks Question" from Maxima and Minima→Maxima and Minima — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let * be a binary operation on Z defined by a * b = a + b - 4 for all a, b ∈ Z.
Show that '*' is both commutative and associative.

If $\text{y}=\log|3\text{x}|,\text{x}\neq0,$ find $\frac{\text{dy}}{\text{dx}}.$

$\text{if}(\tan ^{-1} x)^{2} + (\cot^{-1} x)^{2} = \frac{5{\pi}^{2}}{8} ,\text{then find x}.$

A is known to speak truth $3$ times out of $5$ times. He throws a die and reports that it is one. Find the probability that it is actually one.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}-\text{x}\text{ on }[0,\pi]$

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=\sqrt{\text{x}^3}\ \text{at}\text{ x}=4$

Using determinants, find the area of the triangle whose vertices are $(1, 4), (2, 3)$ and $(-5, -3)$. Are the given points collinear?

A coin is tossed three times. Find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ in each of the following:
A = Heads on third toss,
B = Heads on first two tosses.

Evaluate the following integrals:
$\int\frac{\text{x}^2\text{dx}}{\text{x}^6-\text{a}^6}\text{dx}$

If the matric $\text{A}=\begin{bmatrix}5 & 2&\text{x} \\\text{y} & \text{z}&-3\\4&\text{t}&-7\end{bmatrix}$ is a symmetric matrix, find $x, y, z$ and $t$.