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Find the greatest value of f(x)=cos(xe[x]+7x23x), x[1,)
A. -1
B. 1
C. 0
D. none of these

Answer
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Hint: We first try to define the domain of the given trigonometric function. Then we state the range of the function for any values of t, cost[1,1]. We also try to get one continuous domain of 2π distance. At the end we find one-point x for which it attains the maximum value.

Complete step by step answer:
The main function of the given f(x)=cos(xe[x]+7x23x) is cos function.
Now in a span of 2π, it will take values of [1,1]. Also, for any values of t, cost[1,1].
We just have to check that we can get a continuous domain of 2π distance.
Now we find a range of xe[x]+7x23x. Here [x] is the greatest integer function which means the output is the greatest integer possible less than x.
As x[1,), we can say [x][1,) with only integer values.
We also know for any value of x; the exponential function always gets only positive value.
So, x[1,), we can say e[x][1e,).
So, we can see the function attains at least one continuous domain of 2π distance as it goes towards infinity.
So, we can say the greatest value of f(x)=cos(xe[x]+7x23x) is 1.
We can also prove it by just showing 1 value of x for which f(x)=cos(xe[x]+7x23x)=1.
Let’s take x=0.
We find the value of xe[x]+7x23x. So, xe[x]+7x23x=0.
So, at x=0, cos0=1. We already got a point.

So, the correct answer is “Option B”.

Note: We need to always shoe at least one point which attains the maximum point. As the part xe[x] can have some fixed points due to the factor that e[x] attains only integer value. We have to show that at least one point of x satisfies the equation.
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