\( g(x) \) is the following function in \( x \in \mathbb{R} \) : \[ g(x)=x^{2}+8 x-6 \] Find the value of \( g(-5) \). \[ (-5)^{2}+8(-5)-6=25-40-6=- \] Use calculus to find the value of \( x \) which gives the minimum value of \( g(x) \).

To find the value of \( g(-5) \), let's substitute \( -5 \) into the function \( g(x) \): \[ g(-5) = (-5)^{2} + 8(-5) - 6 = 25 - 40 - 6 = -21 \] So, \( g(-5) = -21 \). Now, to find the value of \( x \) that gives the minimum value of \( g(x) \), we can use calculus. First, we take the derivative of \( g(x) \): \[ g'(x) = 2x + 8 \] Setting the derivative equal to zero to find critical points: \[ 2x + 8 = 0 \implies 2x = -8 \implies x = -4 \] Next, we can confirm that this is a minimum by checking the second derivative: \[ g''(x) = 2 \] Since \( g''(x) = 2 \) is positive, we can conclude that \( g(x) \) attains its minimum value at \( x = -4 \).