If the function f(x) = begin{cases} tan^{-1}x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For a function $f(x)$ to have a local minimum at $x = a$,the value of the function at $x = a$ must be less than or equal to the values of the func

Vedclass · Accepted Answer

$\left( -\infty, \frac{\pi}{4} \right]$

Vedclass · Answer

(C) For a function $f(x)$ to have a local minimum at $x = a$,the value of the function at $x = a$ must be less than or equal to the values of the function in the immediate neighborhood of $a$.Specifically,for $x = 1$,we require $f(1) \le f(1 - h)$ and $f(1) \le f(1 + h)$ for a small positive $h$.First,calculate $f(1)$ using the definition for $x \ge 1$: $f(1) = \sec^{-1}(1) + \lambda = 0 + \lambda = \lambda$.Next,consider the limit as $x 	o 1^-$: $\lim_{x 	o 1^-} f(x) = 	an^{-1}(1) = \frac{\pi}{4}$.For $x = 1$ to be a local minimum,the value at $x = 1$ must be less than or equal to the values just to the left of $1$. Thus,we require $\lambda \le \frac{\pi}{4}$.Also,for $x > 1$,$f(x) = \sec^{-1}(x) + \lambda$. Since $\sec^{-1}(x)$ is an increasing function for $x \ge 1$,$f(x) > f(1)$ for all $x > 1$ regardless of $\lambda$.Therefore,the condition for a local minimum at $x = 1$ is simply $\lambda \le \frac{\pi}{4}$,which corresponds to the interval $\left( -\infty, \frac{\pi}{4} ight]$.

If the function $f(x) = \begin{cases} \tan^{-1}x; & x < 1 \\ \sec^{-1}x + \lambda; & x \ge 1 \end{cases}$ has a local minimum at $x = 1$,then the range of $\lambda$ is:

Similar Questions

On the interval $[0, 1],$ the function $f(x) = {x^{25}}{(1 - x)^{75}}$ takes its maximum value at the point

Find the minimum value of $64 \sec x + 27 \csc x$ for $0 < x < \frac{\pi}{2}$.

The abscissa of the points,where the tangent to the curve $y=x^3-3x^2-9x+5$ is parallel to the $X$-axis,are

$A$ cone of maximum volume is inscribed in a sphere of radius $R$. The ratio of the height of the cone to the diameter of the sphere is:

Given $\lambda \in [0, 20]$,find the number of integral values of $\lambda$ for which the function $f(x) = x^3 - 12x + \lambda$ has a point of local maxima.

Vedclass Test Series

Exam Paper Generator

Online Exam Module