The Second derivative test for Concavity

[22990atinesh]

I understand the 1st derivative test for testing concavity which says

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.

But I'm confused with 2nd derivative test which says

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.

If f'' < 0 or f'' > 0, then it means its a number (negative or positive). Which means f' is linear and function quadratic. Please correct me If I'm wrong.

 

[micromass]

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 $f^{\prime\prime}>0$.

 

[22990atinesh]

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 $f^{\prime\prime}>0$.

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 ?

 

[HallsofIvy]

Yes. You had said initially that

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.

Of course, if f' is increasing, its derivative, f'', is positive and if f' is decreasing, f'' is negative.

 

[Mark44]

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 ?

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.

 

[22990atinesh]

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.

 

[Mark44]

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 derivative (or first derivative) indicates where the function is increasing or decreasing (or zero).

The Second Derivative Test tells us the behaviour of the slope of the function whether it is increasing or decreasing.

Yes, the second derivative tells us whether the derivative is increasing or decreasing or zero.

 

[22990atinesh]

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 want to 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

 

[Mark44]

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.

Yes. The curve could be concave up like y = x² on [0, ∞) or could be concave down like y = √x on [0, ∞).

If we want to 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

Yes.

 

[22990atinesh]

Thanx Mark44 :)