Mathematics 2007 JAMB Past Questions - Question 13: Find the value of x for which the function F(x) = 2 x³ - 4X + 4 has a maximum value

Correct Answer

B

Explanation

d f (x) = 6x² - 2x – 4 =0 when at maximum → (3x +2)(x-1) = 0 → x = 1To find the value of x for which the function F(x) = 2x³ - 4x + 4 has a maximum value, we need to find the critical points of the function. The critical points occur where the derivative of the function is equal to zero or does not exist. Let's find the derivative of F(x):F'(x) = 6x² - 4To find the critical points, we set F'(x) equal to zero and solve for x:6x² - 4 = 0Adding 4 to both sides:6x² = 4Dividing both sides by 6:x² = 4/6Simplifying:x² = 2/3Taking the square root of both sides:x = ±√(2/3)So, we have two critical points: x = √(2/3) and x = -√(2/3).To determine if these critical points correspond to a maximum or minimum, we can analyze the second derivative of F(x).F''(x) = 12xFor a maximum, the second derivative must be negative. Let's evaluate F''(x) at the critical points:F''(√(2/3)) = 12√(2/3) > 0F''(-√(2/3)) = -12√(2/3) < 0Since F''(-√(2/3)) is negative, x = -√(2/3) corresponds to a maximum.Therefore, the value of x for which the function F(x) = 2x³ - 4x + 4 has a maximum value is x = -√(2/3).