Addison Gross

Answered

2022-02-01

A)

B)

C)

D) None of these.

Answer & Explanation

Ydaxq

Expert

2022-02-02Added 12 answers

Let ${e}^{\left|\mathrm{sin}x\right|}=t\in [1,e]$

So equation transformed into${t}^{2}+4at+1=0$

Above equation must have two distinct solution in [1, e]

For two distinct solution we must have$f\left(1\right)\ge 0,f\left(e\right)\ge 0,4{a}^{2}-4>0\text{and}\text{}1-2ae$

where$f\left(t\right)={t}^{2}+4at+1$

So equation transformed into

Above equation must have two distinct solution in [1, e]

For two distinct solution we must have

where

Rosa Nicholson

Expert

2022-02-03Added 13 answers

Say $s=|\mathrm{sin}x|$ , then $s\in [0,1]$ and we need to find for which s has $-4a={e}^{s}+{e}^{-s}$ two solutions. Since function $f(s)={e}^{s}+{e}^{-s}$ is even and increases for $s\ge 0$ we see that ${f}_{max}=f(1)=e+\frac{1}{e}\text{}\text{and}\text{}{f}_{min}=f(0)=2$ . So
$2<-4a\le e+\frac{1}{e}$
or $-\frac{1}{2}>a\ge -\frac{1+{e}^{2}}{2e}$
Since $-\frac{e}{4}>-\frac{1+{e}^{2}}{2e}$ the answer is (A).

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