Problem 60 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The function is \(f(x) = \frac{1}{12}x^4 + 6x + 3\)

Step 1 - Integrate the Second Derivative

The second derivative of \(f\) is given as \(f''(x) = x^2\). To find the first derivative \(f'(x)\), integrate \(f''(x)\). The indefinite integral of \(x^2\) is \(\frac{1}{3}x^3 + C_1\), where \(C_1\) is the constant of integration.

Step 2 - Use Given Condition to Solve for First Constant

The value of \(f'(0)\) is given as \(6\). Substitute \(x = 0\) and \(f'(0) = 6\) into the expression for \(f'(x)\) obtained from the previous step, resulting in \(\frac{1}{3}(0)^3 + C_1 = 6\). Solving this, we find that \(C_1 = 6\). Thus, \(f'(x) = \frac{1}{3}x^3 + 6\).

Step 3 - Integrate the First Derivative

Next, integrate \(f'(x)\) to obtain the original function \(f(x)\). The integral of \(\frac{1}{3}x^3 + 6\) with respect to \(x\) is \(\frac{1}{12}x^4 + 6x + C_2\), where \(C_2\) is another constant of integration.

Step 4 - Use Given Condition to Solve for Second Constant

The value of \(f(0)\) is given as \(3\). Substitute \(x = 0\) and \(f(0) = 3\) into the expression for \(f(x)\) from the previous step. This yields \(\frac{1}{12}(0)^4 + 6(0) + C_2 = 3\). Solving this, we get \(C_2 = 3\). Thus, the original function is \(f(x) = \frac{1}{12}x^4 + 6x + 3\).