Problem 76 - Essential Calculus

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

Step-by-Step Solution

Answer

Therefore, using the information provided in the exercise and considering \(g(x)=-f(x)\), it can be concluded that \(g^{\prime}(0) > 0\).

Step 1: Understanding the Function

The function \(g(x)\) is given as \(g(x)=-f(x)\). So, the derivative of \(g(x)\) is going to be the opposite of the derivative of \(f(x)\), because the derivative of a constant multiplied function will be the constant multiplied by the derivative of the function.

Step 2: Determine the Value of Derivative of Original Function

From the exercise, we know that \(f^{\prime}(x) < 0\) for the interval such that -4 < x < 6, and 0 is within that interval. Therefore, we can say that \(f^{\prime}(0) < 0\).

Step 3: Calculate the Derivative of New Function

For the function \(g(x)\), since \(g(x)=-f(x)\), the derivative \(g^{\prime}(x)\) will be -\(f^{\prime}(x)\). So, \(g^{\prime}(0)\) will be -\(f^{\prime}(0)\). Since \(f^{\prime}(0) < 0\), the value of \(g^{\prime}(0)\) will be positive because -(-) is positive.

Key Concepts

Differentiability

When we say a function is differentiable, it means the function has a derivative at every point in its domain. This characteristic tells us that the function is smooth and continuous without breaks or sharp turns. Differentiability ensures that:

• We can find the slope of the tangent at any point on the curve.
• The function behaves predictably, making calculus tools applicable.

In our exercise, since function \(f\) is differentiable for all \(x\), we know that \(f(x)\) changes smoothly across all intervals. This smooth transition enables us to evaluate its derivative, \(f'(x)\), to understand how the function is behaving and changing direction, which will assist in finding the derivative of \(g(x)\).

Sign of Derivative

The sign of a derivative reveals whether a function is increasing or decreasing. Specifically:

• If \(f'(x) > 0\), the function \(f(x)\) is increasing.
• If \(f'(x) < 0\), the function \(f(x)\) is decreasing.

In the original exercise, the derivative \(f'(x)\) switches signs, indicating changes in the direction of \(f(x)\): - From \((-\infty, -4)\), \(f'(x) > 0\), so \(f(x)\) increases. - From \((-4, 6)\), \(f'(x) < 0\), so \(f(x)\) decreases. - From \((6, \infty)\), \(f'(x) > 0\), so \(f(x)\) increases again. This understanding is key to finding the derivative of \(g(x)\), since \(g'(x) = -f'(x)\). For \(x = 0\), since \(f'(0) < 0\), the derivative \(g'(0)\) becomes positive, leading us to recognize an increase in \(g(x)\) at \(x=0\).

Function Behavior

Function behavior describes how a function changes across its domain. By examining derivatives, we can determine potential turning points, intervals of increase and decrease, and the overall shape of the graph.

• Positive derivatives indicate upward trends.
• Negative derivatives suggest falling trends.

For our functions, the behavior of \(f(x)\) is directly mirrored by its derivative \(f'(x)\). By analyzing \(f'(x)\)'s sign changes, we identify intervals where \(f(x)\) rises or falls. This information is inverted for \(g(x) = -f(x)\), as the negative sign flips the behavior with respect to its derivative. Hence, when \(f(x)\) decreases, \(g(x)\) increases, and vice versa. Understanding these shifts is crucial for predicting how \(g(x)\) behaves, especially around specific values like \(x=0\). In this case, since \(f'(0) < 0\), \(g'(0) > 0\), pointing out that \(g(x)\) is positively trending at \(x=0\).