If the set of all values of a is [alpha, beta | vedclass

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

Question

(A) For $f(x)$ to have its largest value at $x = 1$,we must have $f(1) \geqslant f(x)$ for all $x$ in the domain.Given $f(1) = 5 - (1)^2 = 4$.For $x \

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) For $f(x)$ to have its largest value at $x = 1$,we must have $f(1) \geqslant f(x)$ for all $x$ in the domain.Given $f(1) = 5 - (1)^2 = 4$.For $x \geqslant 1$,$f(x) = 5 - x^2$,which is a decreasing function,so $f(x) \leqslant f(1) = 4$ is satisfied for all $x \geqslant 1$.For $a \leqslant x Since $3x + |a^2 - 4|$ is an increasing function of $x$,its supremum on $[a, 1)$ is $3(1) + |a^2 - 4| = 3 + |a^2 - 4|$.Thus,we require $3 + |a^2 - 4| \leqslant 4$,which implies $|a^2 - 4| \leqslant 1$.This inequality is equivalent to $-1 \leqslant a^2 - 4 \leqslant 1$,which simplifies to $3 \leqslant a^2 \leqslant 5$.Taking the square root,we get $a \in [-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$.Comparing this with $[\alpha, \beta] \cup [\gamma, \delta]$,we have $\alpha = -\sqrt{5}, \beta = -\sqrt{3}, \gamma = \sqrt{3}, \delta = \sqrt{5}$.Therefore,$\alpha + \beta + \gamma + \delta = -\sqrt{5} - \sqrt{3} + \sqrt{3} + \sqrt{5} = 0$.

If the set of all values of $a$ is $[\alpha, \beta] \cup [\gamma, \delta]$ for which the function $f(x) = \begin{cases} 3x + |a^2 - 4|; & a \leqslant x < 1 \\ 5 - x^2; & x \geqslant 1 \end{cases}$ has its largest value at $x = 1$,then find the value of $(\alpha + \beta + \gamma + \delta)$.

Similar Questions

Let $f : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathbb{R}$ be defined by $f(x) = (\log(\sec x + \tan x))^3$. Then:

Let $f, g: R \rightarrow R$ be defined,respectively,by $f(x) = x + 1$ and $g(x) = 2x - 3$. Find $f+g$,$f-g$,and $\frac{f}{g}$.

Let $g(x) = 1 + x - [x]$ and $f(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0 \end{cases}$,where $[x]$ denotes the greatest integer less than or equal to $x$. Then for all $x$,$f(g(x)) = $

Which one of the following is not bounded on the intervals as indicated?

Vedclass Test Series

Exam Paper Generator

Online Exam Module