Question

The maximum value of $4 \, \sin^2 \, x - 12 \sin \, x + 7$ is

• A. 25
• B. 4
• C. does not exist
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

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

To find the maximum value of the given expression 4sin2 x −12sin x +7, we can use calculus. Let's differentiate the expression with respect to x to find its critical points.

Given expression: f(x)=4sin2 x −12sin x +7

Let's find the derivative of f ′(x)=dxd ​(4sin2 x −12sin x +7) f ′(x)=8cos2 x −12cos x

To find critical points, we set f ′(x) equal to 0 and solve for x : 8cos2 x −12cos x =0

Dividing both sides by 4: 2cos2 x −3cos x =0

Now, we can use the trigonometric identity cos2 x =2cos2 x −1 to substitute for cos2 x :

2(2cos⁡2x−1)−3cos⁡x=0

4cos2 x −2−3cos x =0

4cos2 x −3cos x −2=0

Let u =cos x , then the equation becomes: 4 u 2−3 u −2=0

Now we can factor this quadratic equation: (4 u +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 cos x is not achievable within the range of the cosine function.

Since we cannot find a real x that satisfies cos x =−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."