Question

The figure shows graphs of f, f ', f '', and f '''. Identify each curve.

Answer 1

f --> d

f' --> c

f'' --> b

f''' --> a

Explanation:

The derivative of a function
`f(x)` is another function
`f'(x)` which values are the slope of the line tangent to the curve
`f(x)` at the point
`x`.

So the derivative at some point will be:

• Positive if the line tangent at the curve at that point has a positive angle with the horizontal axis (this means that curve is increasing from left to right).

• Negative if the line tangent at the curve at that point has a negative angle with the horizontal axis (this means that the curve is decreasing from left to right).

• Zero if the line tangent at the curve at that point is parallel to the horizontal axis (this means that the curve is remaining constant, has an inflection point or has reached a local maximum or minimum point).

The
`<strong>d</strong>` curve has a steady decreasing rate when
`x` is close zero and has one local maximum and one local minimum where the slope of the tangent line should be zero. The
`<strong>c</strong>` curve shows this behaviour so we can say that
`c` is the derivative of
`<strong>d</strong>`.

The
`c` curve reach a minimum point when
`x=0` so the derivative should be zero at that point, with a decrasing rate for
`x<0` (negative derivative) and an increasing rate for
`x>0` (positive derivative). The
`<strong>b</strong>` curve shows this behaviour so we can say that
`b` is the derivative of
`c`.

The
`b` curve is constantly increasing, so its derivative is going to be always positive and has an inflection point in
`x=0` where the slope of the tangent line should be zero, then
`<strong>a</strong>` is the derivative of
`<strong>b</strong>`

Recapitulating:

• 
  `c` is the derivative of
  `d`
• 
  `b` is the derivative of
  `c`
• 
  `a` is the derivative of
  `b`

Then:

f --> d

f' --> c

f'' --> b

f''' --> a