A function f(x) passes through the origin and its first derivative is 3x + 2. What is f(x)?

  • A \(y= \frac{{3x^2}}{{2}} + 2x\)
  • B \y = \frac{{3x^2}}{2} + x\)
  • C \(y = 3x^2 + \frac{x}{2}\)
  • D \((y = 3x^2 + 2x)\)
jamb 2000mathematics

The correct answer is A. \(y= \frac{{3x^2}}{{2}} + 2x\)

The solution to this problem is as follows:

Since the first derivative of \(f(x)\) is \(3x + 2\), we can find \(f(x)\) by integrating the first derivative with respect to \(x\). This gives us:

\(f(x) = \int (3x + 2) dx = \frac{3x^2}{2} + 2x + C\)

where \(C\) is the constant of integration. Since the function passes through the origin, we have \(f(0) = 0\). Plugging this into the above equation, we get:

\(0 = \frac{3 \cdot 0^2}{2} + 2 \cdot 0 + C \Rightarrow C = 0\)

Therefore, \(f(x) = \frac{3x^2}{2} + 2x\).

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