If α and β are such that the function f(x) defined by fx=αx2-β, for |x|<1-1|x|, for |x|≥1 is differentiable everywhere then the ordered pair (α,β)=

AP EAMCET · Maths · Continuity and Differentiability

If $α$ and $β$ are such that the function $f (x)$ defined by $f (x) = \{\begin{cases} α x^{2} - β, for | x | < 1 \\ \frac{- 1}{| x |}, for | x | \geq 1 \end{cases}$ is differentiable everywhere then the ordered pair $(α, β) =$

A $(- \frac{1}{2}, - \frac{3}{2})$
B $(\frac{1}{2}, - \frac{3}{2})$
C $(\frac{1}{2}, \frac{3}{2})$
D $(- \frac{1}{2}, \frac{3}{2})$

Verified Solution

Answer & Solution

Correct Answer

(C) $(\frac{1}{2}, \frac{3}{2})$

Step-by-step Solution

Detailed explanation

It is given that the function is differentiable at all points, By the first principle, LHDat x=1=RHDat x=1 limx→1-f(x)-f(1)x-1=limx→1+f(x)-f(1)x-1 limx→1-αx2-β-α+βx-1=limx→1+-1x+1x-1…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From AP EAMCET

Match the following.

List - $I$ Reaction		List - $II$ Catalyst
$(A)$	$N_{2} (g) + 3 H_{2} (g) ⟶ 2 {NH}_{3} (g)$	$(I)$	$Ni$
$(B)$	$2 H_{2} (g) + O_{2} (g) ⟶ 2 H_{2} O (l)$	$(II)$	$Pt$
$(C)$	$CO (g) + 3 H_{2} (g) ⟶ {CH}_{4} (g) + H_{2} O (g)$	$(III)$	$ZnO - {Cr}_{2} O_{3}$
$(D)$	$CO (g) + 2 H_{2} (g) ⟶ {CH}_{3} OH (g)$	$(IV)$	$Fe$

The correct answer is:

AP EAMCET 2019 Easy

If $A$ and $B$ are two independent events such that $P(B)=\frac{2}{7}, P\left(A \cup B^c\right)=0.8$, then $P(A)$ is equal to :

AP EAMCET 2006 Medium

If $\cos \alpha+\cos \beta=a$ and $\sin \alpha+\sin \beta=b$, then match the items given in List $-A$ with those of their values in List-B

List-A	Liat-B
(I) $\tan \left(\frac{\alpha+\beta}{2}\right)=$	(a) $\frac{ b }{ a }$
(II) $\cos (\alpha+\beta)=$	(b) $\frac{2 a b}{a^2+b^2}$
(III) $\sin (\alpha+\beta)^{-}$	(c) $\frac{2 a b}{a^2-b^2}$
(IV) $\tan (\alpha+\beta)=$	(d) $\frac{a^2-b^2}{a^2+b^2}$
	(e) $\frac{a^2+b^2}{a^2-b^2}$

AP EAMCET 2023 Medium

The number of subsets of $\{1,2,3, \ldots, 9\}$ containing at least one odd number is

AP EAMCET 2009 Medium

Explore more questions on app

List-A	Liat-B
(I) \(\tan \left(\frac{\alpha+\beta}{2}\right)=\)	(a) \(\frac{ b }{ a }\)
(II) \(\cos (\alpha+\beta)=\)	(b) \(\frac{2 a b}{a^2+b^2}\)
(III) \(\sin (\alpha+\beta)^{-}\)	(c) \(\frac{2 a b}{a^2-b^2}\)
(IV) \(\tan (\alpha+\beta)=\)	(d) \(\frac{a^2-b^2}{a^2+b^2}\)
	(e) \(\frac{a^2+b^2}{a^2-b^2}\)