Question

Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=f(x)+5 \quad g^{\prime}(0) $$

Short answer

The derivative of function \(g\) at \(x = 0\) which is \(g'(0)\) is less than zero, i.e., \(g'(0) < 0\).

Step-by-step solution

Understand the given derivative conditions

The problem states that the derivative of \(f\) \(f'(x)\) is positive in intervals \((-\infty, -4)\) and \((6, \infty)\) and negative on \((-4, 6)\). The inequality sign changes at -4 and 6, which indicates that the function \(f\) has a local maximum at \(x = -4\) and a minimum at \(x = 6\).

Analyze the function \(g(x) = f(x) + 5\)

The function \(g(x) = f(x) + 5\) is simply the function \(f(x)\) shifted vertically upwards by 5 units. This shift does not affect the derivative, as the derivative of a constant is zero. Therefore, \(g'(x) = f'(x)\).

Find \(g'(0)\)

Now, you want to find the value of \(g'(0)\). We know that \(g'(x) = f'(x)\), so \(g'(0) = f'(0)\). From our information about the derivative \(f'(x)\), we know that \(f'(x)<0\) on \(-4 < x < 6\). The number 0 falls within this interval, thus \(f'(0) < 0\) and consequently \(g'(0) < 0\).