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

Hint:

For quadratic functions, if the vertex lies outside the domain, evaluate the function at the endpoints of the domain to find the maximum or minimum value.

Answer 1

The Correct Option is D

We are asked to find the maximum value of the function: \[ f(x) = 4 \sin^2 x - 12 \sin x + 7. \] Step 1: Substituting \( y = \sin x \).
Let \( y = \sin x \), where \( y \in [-1, 1] \) because \( \sin x \) takes values between -1 and 1. Now, the function becomes: \[ f(y) = 4y^2 - 12y + 7. \] This is a quadratic function in \( y \), and we need to find its maximum value for \( y \in [-1, 1] \).
Step 2: Finding the vertex of the quadratic function.
The general form of a quadratic function is \( ay^2 + by + c \), and the vertex occurs at: \[ y = -\frac{b}{2a}. \] For the function \( f(y) = 4y^2 - 12y + 7 \), we have: \[ a = 4, \quad b = -12, \quad c = 7. \] So, the vertex occurs at: \[ y = -\frac{-12}{2 \times 4} = \frac{12}{8} = 1.5. \]
Step 3: Checking if the vertex is within the domain.
Since the vertex occurs at \( y = 1.5 \), which is outside the domain \( [-1, 1] \), we need to check the function values at the endpoints of the domain.
Step 4: Evaluating the function at the endpoints.
First, evaluate \( f(y) \) at \( y = 1 \): \[ f(1) = 4(1)^2 - 12(1) + 7 = 4 - 12 + 7 = -1. \] Next, evaluate \( f(y) \) at \( y = -1 \): \[ f(-1) = 4(-1)^2 - 12(-1) + 7 = 4 + 12 + 7 = 23. \]
Step 5: Conclusion.
The maximum value of the function occurs at \( y = -1 \), and the value is \( 23 \). Therefore, the correct answer is (d) None of these.