Answer is: option3

Solution:

The derivative of a function is zero at its turning points (local maxima or minima).

So, if a graph has turning points at certain \( x \)-values, then its derivative graph will cross the x-axis at those same \( x \)-values.

Curve III has two turning points.

Curve I crosses the \( x \)-axis at the same \( x \)-values where Curve III has turning points.
  • So, Curve I is the derivative of Curve III
  → That means: \( f = \text{Curve III} \) and \( f' = \text{Curve I} \)

Similarly, Curve I has two turning points.

Curve II crosses the \( x \)-axis at the same \( x \)-values where Curve I has turning points.
  • So, Curve II is the derivative of Curve I
  → That means: \( f'' = \text{Curve II} \)

Therefore:

1. \( f = \text{Curve III} \)
2. \( f' = \text{Curve I} \)
3. \( f'' = \text{Curve II} \)

Correct answer: (C)