The function f(x)= x 2 + 2 x has a local minimum at

Step-by-step solution

Given the function f(x) = x 2 + 2 x . To find the local extrema, calculate the first derivative f'(x) : f'(x) = d dx ( x 2 + 2x⁻¹ ) = 1 2 - 2x⁻² = 1 2 - 2 x^2 . Set f'(x) = 0 to find the critical points: 1 2 - 2 x^2 = 0 1 2 = 2 x^2 x^2 = 4 x = 2 . Calculate the second derivative f''(x) to determine the nature of the critical points: f''(x) = d dx ( 1 2 - 2x⁻² ) = 0 - 2(-2)x⁻³ = 4 x^3 . Evaluate f''(x) at the critical points: For x = 2 , f''(2) = 4 2^3 = 4 8 = 1 2 > 0 . Since f''(2) > 0 , the function has a local mi