Download App

STD 12 - 6. Application of derivatives — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 6. Application of derivatives4 Marks
MCQ
Let $f(x) = \int\limits_0^x {{{\cos t} \over t}dt,\,\,x > 0} $ then $f(x)$ has
  • A
    Maxima when $n = - 2,\, - 4,\, - 6,\,.....$
  • ✓
    Maxima when $n = - 1,\, - 3,\, - 5,\,....$
  • C
    Minima when $n = 0,\,2,\,4,....$
  • D
    None of these
✓

Answer

Correct option: B.
Maxima when $n = - 1,\, - 3,\, - 5,\,....$
b
(b) $f(x) = \int\limits_0^x {\frac{{\cos t}}{t}} \,dt$, $x > 0$

==> $f'(x) = \frac{{\cos x}}{x}$, $x > 0$

==>$f'(x) = 0 \Rightarrow \frac{{\cos x}}{x} = 0$

==> $x = (2n + 1){\rm{ }}\frac{\pi }{{\rm{2}}}$, for $n \in z$.

Now $f''(x) = \frac{{ - x\sin x - \cos x}}{{{x^2}}}$

$\therefore$ $f''[(2n + 1)\pi /2] = \frac{{ - 2}}{{(2n + 1)\pi }}\,{( - 1)^n}$ = $\frac{{2{{( - 1)}^{n + 1}}}}{{(2n + 1)\pi }}$.

Thus $f''(x) > 0$$n = - 2,\, - 4,\, - 6,........$

$f''(x) < 0$ $n = 0,\,2,\,4,\,........$

$f''(x) > 0$ $n = 1,\,3,\,5........$

$f''(x) < 0$ $n = - 1,\, - 3,\, - 5........$

Thus $f(x)$ attain maximum for $n = - 1,\, - 3,\, - 5$,…. and minimum for $n = 1,\,3,\,5$,…..

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $A$ and $B$ are square matrices of same order which $AB = A$, $BA = B$, then $(A + I)^5$ is equal to (where $I$ is unit matrix)
→
If $A$ and $B$ are two independent events, then $A$ and $\bar B$ are
→
Let $f(x) = log_e\,(sin\,x),$ $(0\,<\,x\,< \pi )$ and $g(x) = sin^{-1}\,(e^{-x}),$ $(x\, \ge \,0)$. If $\alpha $ is a positive real number such that $a$ $ = (fog)’(\alpha )$ and $b = (fog)(\alpha ),$ then
→
The chord $ PQ $ of the rectangular hyperbola $xy = a^2$  meets the axis of $x$ at $A ; C $ is the mid point of $  PQ\ \& 'O' $ is the origin. Then the $ \Delta ACO$  is :
→
If $\frac{{d[f(x)]}}{{dx}} = g(x)$ for $a \le x \le b,$ then $\int_a^b {f(x)\,\,g(x)\,dx} $ equals
→
Let $M=\left\{A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right): a, b, c, d \in\{\pm 3, \pm 2, \pm 1,0\}\right\} .$ Define $f: M \rightarrow z$, as $f(A)=\operatorname{det}(A)$ for all $A \in M$, where $Z$ is set of all integers. Then the number of $A \in M$ such that $f(A)=15$ is equal to $.....$
→
$\int_{\,0}^{\,1} {\,\sin \left( {2{{\tan }^{ - 1}}\sqrt {\frac{{1 + x}}{{1 - x}}} } \right)\,dx = } $
→
The lengths of the sides and the diagonal of an isosceles trapezium form a two-element set $\{a, b\}$. If $a > b$, then $a / b$ equals
→
The sum of all values of $\theta \in[0,2 \pi]$ satisfying $2 \sin ^2 \theta=\cos 2 \theta$ and $2 \cos ^2 \theta=3 \sin \theta$ is
→
${I_n} = \int\limits_0^{\frac{\pi }{4}} {{{\tan }^n}x\,dx} $ then $\mathop {\lim }\limits_{n \to \infty } \,\,n({I_n} + {I_{n - 2}})$ equals
→