The minimum value of function f(x)=2 x+x in interval [0, 2 ] : - Vidyadip

MCQ

The minimum value of function $f(x)=2 \cos x+x$ in interval $\left[0, \frac{\pi}{2}\right]$ :

A
2
B
$\frac{\pi}{6}+\sqrt{3}$
✓
$\frac{\pi}{2}$
D
None of these

✓

Answer

Correct option: C.

$\frac{\pi}{2}$

(C)$f(x)=2 \cos x+x \Rightarrow f^{\prime}(x)=-2 \sin x+1$
now, $f^{\prime}(x)=0 \Rightarrow-2 \sin x+1=0$
$\Rightarrow \quad \sin x=\frac{1}{2}=\sin \frac{\pi}{6} \Rightarrow x=\frac{\pi}{6}$
$\Rightarrow \quad x=\frac{\pi}{6} \in\left[0, \frac{\pi}{2}\right]$
$\therefore \quad f(0)=2+0=2, f\left(\frac{\pi}{6}\right)=2 \cos \frac{\pi}{6}+\frac{\pi}{6}$
$=2\left(\frac{\sqrt{3}}{2}\right)+\frac{\pi}{6}=\sqrt{3}+\frac{\pi}{6}$
and $\quad f\left(\frac{\pi}{2}\right)=0+\frac{\pi}{2}=\frac{\pi}{2}$
The minimum value of $f(x)$ is $\frac{\pi}{2}$ at $x=\frac{\pi}{2}$.

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 "M.C.Q (1 Marks)" from Applications of Derivative→Applications of Derivative — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $A=\{1,2,3\}$. Then number of equivalence relations containing $(1,2)$ is __________ .

The distance between the planes 2x + 2y - z +2 = 0 and 4x + 4y - 2z + 5 = 0 is:

$\frac{1}{2}$
$\frac{1}{4}$
$\frac{1}{6}$
$\text{None of these}$

Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Let $R$ be a relation defined on $A \times B$ such that $R =$ $\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right): a_1 \leq b_2\right.$ and $\left.b_1 \leq a_2\right\}$. Then the number of elements in the set $R$ is

$\sin \left(\tan ^{-1} x\right)$, where $|x|<1$, is equal to

Choose the correct answer from the given four options.

The reflection of the point $(\alpha,\beta,\gamma)$ in the xy–plane is:

$(\alpha,\beta,0)$
$(0,0,\gamma)$
$(-\alpha,-\beta,\gamma)$
$(\alpha,\beta,-\gamma)$

The minimum value of ${e^{(2{x^2} - 2x + 1){{\sin }^2}x}}$ is

Let $k$ and $K$ be the minimum and the maximum values of the function $f\left( x \right) = \frac{{{{\left( {1 + x} \right)}^{0.6}}}}{{1 + {x^{0.6}}}}$ in $[0, 1 ]$ respectively, then the ordered pair $(k, K)$ is equal to

If $f(x) = \left\{ \begin{array}{l}x,\;\;{\rm{when\,\,}}0 < x < 1/2\\1,\;\;\;{\rm{when\,\, }}x = 1/2\\1 - x,{\rm{when}}\;{\rm{1/2}} < x < {\rm{1}}\end{array} \right.$, then

The integral $\int{ \cfrac{d x}{(x+4)^{\frac{8}{7}}(x-3)^{\frac{6}{7}}}}$ is equal to

(where $\mathrm{C}$ is a constant of integration)

If the system of equations

$x+y+a z=b$

$2 x+5 y+2 z=6$

$x+2 y+3 z=3$

has infinitely many solutions, then $2 a+3 b$ is equal to $...........$.