Question

The function $ f(x) = 2 + 4x^2 + 6x^4 + 8x^6 $ has

• A. Only one maxima
• B. Only one minima
• C. No maxima and minima
• D. Many maxima and minima

Hint:

When analyzing polynomials with even powers, always check for symmetry and use the second derivative test to identify minima or maxima.

Answer 1

The Correct Option is B

Step 1: Find the derivative of \( f(x) \)
The first derivative of \( f(x) \) is: \[ f'(x) = 8x + 24x^3 + 48x^5 \]
Step 2: Set the derivative equal to zero to find critical points \[ 8x + 24x^3 + 48x^5 = 0 \] Factor out \( 8x \): \[ 8x(1 + 3x^2 + 6x^4) = 0 \] Thus, \( x = 0 \) is a critical point.
The quadratic \( 1 + 3x^2 + 6x^4 = 0 \) has no real solutions.
Step 3: Analyze the function
Since the function is composed of even powers of \( x \), it has symmetry and no local maxima.
The only critical point is \( x = 0 \), and evaluating the second derivative at \( x = 0 \), we find that \( f(x) \) has a local minimum at this point.
Step 4: Conclusion
Thus, the function has only one minima.