Question

If the possible values of ‘a’ such that the inequality 3 − x² > |x − a| has at least one negative solution is a ∈ (-13/4, λ) then find λ.

Answer 1

See Fig.. If a < a_min, then the curve y = |x − a| will not intersect the curve y = −x² + 3.

Similarly, if a > a_max, then the curve y = |x − a| will not intersect the curve y = 3 − x² for any x ∈ (−√3, 0).

Case Ι: 

L₁ is tangent to y = − x² + 3 and its equation is y = x − a 

Therefore,

dy/dx = −2x = 1

That is, 

x = − 1/2

Therefore, (-1/2,  11/4) lies on y = x − a

Thus, point of contact is (-1/2,  11/4).

Since, it lies on y = x − a.

Therefore a_min = − 13/4

So, the inequality has a negative solution if − 13/4 < a < 0. (1) 

Case ΙΙ:

Line L₂ is y = a − x and passes through (0, 3) if a = 3.

Thus, the inequality has a negative solution if − 3 < a < 3. (2) 

From Eqs. (1) and (2), we get that the inequation has at least one negative solution if − 13/4 < a < 3.