Question

The minimum value of \( f(x) = 7x^4 + 28x^3 + 31 \) is:

• A. 12
• B. 10
• C. 38
• D. 76
• E. 56

Hint:

For polynomial functions, find critical points by setting the first derivative to zero, then evaluate function values to determine minima or maxima.

Answer 1

The Correct Option is B

To find the minimum value of the function, differentiate \( f(x) \): \[ f'(x) = 28x^3 + 84x^2 \] Factorizing: \[ f'(x) = 28x^2 (x + 3) \] Setting \( f'(x) = 0 \): \[ 28x^2 (x + 3) = 0 \] This gives the critical points: \[ x = 0 \quad \text{or} \quad x = -3 \] Substituting these values into \( f(x) \): \[ f(0) = 7(0)^4 + 28(0)^3 + 31 = 31 \] \[ f(-3) = 7(-3)^4 + 28(-3)^3 + 31 = 567 - 756 + 31 = 10 \] Thus, the minimum value of \( f(x) \) is 10.