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The Second derivative test for Concavity

  1. Apr 2, 2014 #1
    The graph of a differentiable function y=f(x) is

    1. concave up on an interval I if f' is increasing on I.
    2. concave down on an interval I if f' is decreasing on I.

    Let y=f(x) is twice differentiable on an interval I
    1. If f'' > 0 on I, the graph of f over I is concave up.
    2. If f'' < 0 on I, the graph of f over I is concave down.

     
  2. jcsd
  3. Apr 2, 2014 #2

    micromass

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    The notation ##f^{\prime\prime}>0## is shorthand for ##f^{\prime\prime}(x) >0## for each ##x## in the domain.

    For example, if ##f(x) = x^4##, then ##f^{\prime\prime}(x) = 12x^2>0## for each ##x##. Thus we denote this by the notation ##f^{\prime\prime}>0##.
     
  4. Apr 2, 2014 #3
    micromass you mean ##f^{\prime\prime} > 0## (or ##f^{\prime\prime}(x) >0##) intuitively means tangent slope at each point on the derivative curve should be positive or negative. Is that correct ?
     
  5. Apr 2, 2014 #4

    HallsofIvy

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    Yes. You had said initially that
    Of course, if f' is increasing, its derivative, f'', is positive and if f' is decreasing, f'' is negative.
     
  6. Apr 2, 2014 #5

    Mark44

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    No.
    If f' > 0 for all x in some interval, then the slope of the tangent is positive on that interval. f'', the second derivative, gives the rate of change of the slope of the tangent.

    As a simple example, let f(x) = x2. Then f'(x) = 2x, and f''(x) = 2.

    Here, f'' > 0 for all real numbers, but the slope of the tangent to this curve is negative when x < 0, and is positive when x > 0.

    What is happening is that the slope of the tangent line to the curve is increasing (from very negative to very positive) as we move from the left to the right.
     
  7. Apr 3, 2014 #6
    Thanx Everybody, My doubt is clear now. Just like First derivative Test tells us the behaviour of the function whether it is increasing or decreasing. The Second Derivative Test tells us the behaviour of the slope of the function whether it is increasing or decreasing.
     
  8. Apr 3, 2014 #7

    Mark44

    Staff: Mentor

    The derivative (or first derivative) indicates where the function is increasing or decreasing (or zero).
    Yes, the second derivative tells us whether the derivative is increasing or decreasing or zero.
     
  9. Apr 4, 2014 #8
    Mark44, One more thing

    Suppose we have given that f'>0 on I => f increases on I, But it doesn't say anything about the curvature of f on I. It just says f is moving upwards as x increases.

    If we wanna know whether f is curved on I or is a staright line we need f'' on I. If f''>0 or f''<0 on I then f is curved on I (Cocave Up or Concae Down), But if f''=0 on I then f is a straight line on I.

    Is above analogy is right
     
  10. Apr 4, 2014 #9

    Mark44

    Staff: Mentor

    Yes. The curve could be concave up like y = x2 on [0, ∞) or could be concave down like y = √x on [0, ∞).
    Yes.
     
  11. Apr 4, 2014 #10
    Thanx Mark44 :)
     
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