Consider the curve (4/cos x)+(1/1 - cos â¡ x) where x∈ (- (π /2) , 0)∪(0 , (π /2)) . The value of a, a∈ R for which the line y=a and the given curve has only one solution is:

Question

Consider the curve (4/cos x)+(1/1 - cos â¡ x) where x∈ (- (π /2) , 0)∪(0 , (π /2)) . The value of a, a∈ R for which the line y=a and the given curve has only one solution is: (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let t=cos x∈ (0 , 1) for x∈ (- (π /2) , 0)∪(0 , (π /2)) then f(t)=(4/t)+(1/1 - t),t∈ (0 , 1) f1(t)=((2 - t) (2 t - 3)/t2 (1 - t)2)>0 for (2/3) < t < 1 < 0 for 0 < t < (2/3) ∴ At t=(2/3),f(t) has global minimum value = 9 ∴ Least value of a for which f(t)=a has solution is 9

Q. Consider the curve cosx4​+1−cos⁡x1​wherex∈(−2π​,0)∪(0,2π​) . The value of a,a∈R for which the line y=a and the given curve has only one solution is:

Let t=cosx∈(0,1) for x∈(−2π​,0)∪(0,2π​) then f(t)=t4​+1−t1​,t∈(0,1) f1(t)=t2(1−t)2(2−t)(2t−3)​>0 for 32​<t<1 <0 for 0<t<32​ ∴ At t=32​,f(t) has global minimum value =9 ∴ Least value of a for which f(t)=a has solution is 9

Q. Consider the curve $\frac{4}{cos x} + \frac{1}{1 - cos x} w h ere x \in (- \frac{π}{2}, 0) \cup (0, \frac{π}{2})$ . The value of $a, a \in R$ for which the line $y = a$ and the given curve has only one solution is: