Question

Find a function \(f\) that satisfies the conditions. $$ f^{\prime \prime}(x)=x^{-3 / 2}, \quad f^{\prime}(1)=2, \quad f(9)=-4 $$

Short answer

The function that satisfies the provided conditions and its second derivative is \(f(x) = 4x^{1/2} + 2x - 22\).

Step-by-step solution

First Integration

Start by computing the first integral of \(f^{\prime \prime}(x)=x^{-3 / 2}\). The antiderivative of \(x^{-3 / 2}\) is \(-2x^{-1/2}\), so we have \(f^{\prime}(x) = -2x^{-1/2} + C\), where \(C\) is the constant of integration.

Calculating the Constant in Step 1

To calculate the constant \(C\), use the given condition \(f^{\prime}(1) = 2\). When we set \(x = 1\) in \(f^{\prime}(x) = -2x^{-1/2} + C\), we get \(C = 2\). Therefore, the first derivative of \(f\) is \(f^{\prime}(x) = -2x^{-1/2} + 2\).

Second Integration

Next, compute the second integral using the derivative \(f^{\prime}(x) = -2x^{-1/2} + 2\). The antiderivative of \(-2x^{-1/2}\) is \(4x^{1/2}\) and of \(2\) is \(2x\). Therefore, we get \(f(x) = 4x^{1/2} + 2x + D\), where \(D\) is another constant of integration.

Calculating the Constant in Step 3

To calculate the constant \(D\), use the given condition \(f(9) = -4\). Setting \(x = 9\) in \(f(x) = 4x^{1/2} + 2x + D\), we get \(D = -22\).