Application Of Derivatives

The correct option is(D): None of these.

To find the maximum value of the given expression 4sin2x−12sinx+7, we can use calculus. Let's differentiate the expression with respect to x to find its critical points.

Given expression: f(x)=4sin2x−12sinx+7

Let's find the derivative of f′(x)=dxd​(4sin2x−12sinx+7) f′(x)=8cos2x−12cosx

To find critical points, we set f′(x) equal to 0 and solve for x: 8cos2x−12cosx=0

Dividing both sides by 4: 2cos2x−3cosx=0

Now, we can use the trigonometric identity cos2x=2cos2x−1 to substitute for cos2x:

2(2cos⁡2x−1)−3cos⁡x=0

4cos2x−2−3cosx=0

4cos2x−3cosx−2=0

Let u=cosx, then the equation becomes: 4u2−3u−2=0

Now we can factor this quadratic equation: (4u+1)(u−2)=0

This gives us two possible solutions: u=−41​ or u=2.

However, the cosine function's range is [−1,1][−1,1], so the value of u (cosine) cannot be 2. Therefore, we only have u=−41​, which implies that x=−¼​. But this value of cosx is not achievable within the range of the cosine function.

Since we cannot find a real x that satisfies cosx=−¼, there are no critical points for the function f(x).

Without any critical points, we can conclude that there are no local maximum or minimum points, which means that the function doesn't have a maximum value. Therefore, the correct option is "None of these."