Problem 61 - College Algebra and Calculus: An Applied Approach

Question

Find a function \(f\) that satisfies the conditions. $$ f^{\prime \prime}(x)=x^{-2 / 3}, \quad f^{\prime}(8)=6, \quad f(0)=0 $$

Step-by-Step Solution

Answer

The function that satisfies the given conditions is \(f(x) = \frac{9}{5}x^{\frac{5}{3}} / 2.

Step 1: Integrate the Second Derivative to Find the First Derivative

Start with the given equation, which is the second derivative of the function \(f\). Integrate the equation \(f^{\prime \prime}(x)=x^{-2 / 3}\) to find \(f^{\prime}(x)\). The indefinite integral of \(x^{-2 / 3}\) with respect to \(x\) is \(3x^{\frac{1}{3}}/2 + C1\), where \(C1\) is the constant of integration.

Step 2: Use the Initial Condition to Determine the Constant C1

Plug \(x=8\) and \(f^{\prime}(8)=6\) into our equation for the first derivative to find the constant \(C1\). Solving for \(C1\), we get \(C1 = 6 - 3(2) = 0\). So, \(f^{\prime}(x) = 3x^{\frac{1}{3}}/2.

Step 3: Integrate to Find the Function Itself

Next, integrate our first derivative function, \(f^{\prime}(x) = 3x^{\frac{1}{3}}/2, to find \(f(x)\). The integral is \(f(x) = \frac{9}{5}x^{\frac{5}{3}} / 2 + C2\), where \(C2\) is another constant of integration.

Step 4: Use the Other Initial Condition to Determine the Constant C2

Use the other initial condition \(f(0)=0\) to find the constant \(C2\). Plugging \(x=0\) and \(f(0)=0\) into our function and solving for \(C2\), we get \(C2 = 0\). Therefore, our function \(f(x)\) is \(f(x) = \frac{9}{5}x^{\frac{5}{3}} / 2.