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 \(c\). $$ g(x)=-f(x) \quad g^{\prime}(-6) \quad 0 $$

Short answer

The appropriate inequality for the given value of \(c\) is \(g' (-6) < 0\).

Step-by-step solution

Relationship of derivatives

First, we need to understand how the derivative of \(g(x) = -f(x)\) relates to the derivative of \(f(x)\). By using the properties of differentiation, we know that when we negate a function, its derivative also gets negated. Hence the derivative function \(g'(x)\) of the function \(g(x)\) would be the negation of the function \(f'(x)\). So, \(g'(x) = -f'(x)\).

Evaluate derivative of \(g(x)\) at \(x = -6\)

Now we need to evaluate the sign of the derivative of \(g(x)\) at \(x = -6\). As \(f'(x)>0\) for \(x \in (-\infty, -4)\), it includes the point \(x = -6\). But \(g'(x) = -f'(x)\), therefore \(g'(-6) < 0\).

Formulate inequality

Since we have found that \(g'(-6) < 0\), the appropriate inequality for the given value of \(c\) is \(g' (-6) < 0\), as per the signs of \(g'(x)\).