why does the second derivative of a function needs to be used to find if a point is local or global maxima?

Question

I understand why the first derivative should be zero to find out if a point is a local maxima/minima..... but can someone explain in layman's terms as to why the second derivative will help us to decide if a function is local/global maxima/minima.... what is the geometric explanation ?

Answer 1

The first derivative is the rate of change of the function. For this you are only concerned with the direction.

f'(x) > 0 if f(x) is going up
f'(x) < 0 if f(x) is going down

Assuming that f(x) is not a constant (which would make this trivial), then if f'(x) = 0 at x then f'(x±δ) ≠ 0 right before and right after x.

You have two possibilities.

f'(x-δ) < 0 < f'(x+δ)

f(x) is going down, flattens, then it starts going up -> f''(x)>0 since the rate f'(x) is going down must be declining for this to happen (the second derivative)

f'(x-δ) > 0 > f'(x+δ)

f(x) is going up, flattens, then it starts going down -> f''(x)<0 since the rate f'(x) is going up must be declining for this to happen (the second derivative).

There is also the third alternative. f(x) goes up, flattens then goes up again or f(x) goes down, flattens, then goes down again. These are points of inflection.

Answer 2

Suppose we have a differentiable function f, and some particular point x where f'(x) = 0 and f''(x) > 0.

Think about what f''(x) > 0 means. The second derivative f'' is the rate of change of the first derivative, f'. So if f''(x) > 0, then the first derivative f' must be increasing at x.

Since we also know f'(x) = 0, then if f' is increasing at x, then we must have f'(x) < 0 to the left of x and f'(x) > 0 to the right of x.

But this means that f is decreasing when we're to the left of x and increasing when we're to the right of x. So f(x) must be the smallest that f gets in that area. So x is a local minimum.

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Now suppose we have a differentiable function f, and some particular point x where f'(x) = 0 and f''(x) < 0.

Think about what f''(x) < 0 means. The second derivative f'' is the rate of change of the first derivative, f'. So if f''(x) < 0, then the first derivative f' must be decreasing at x.

Since we also know f'(x) = 0, then if f' is decreasing at x, then we must have f'(x) > 0 to the left of x and f'(x) < 0 to the right of x.

But this means that f is increasing when we're to the left of x and decreasing when we're to the right of x. So f(x) must be the biggest that f gets in that area. So x is a local maximum.

Answer 3

The second derivative will tell you if the slope is increasing or decreasing. If you draw smooth curves with minimums and maximums, you can convince youself that at a minimum the slope mut be increasing (or at least not decreasing), and at a maximum the slope must be decreasing (or at least not increasing).

Another explanation has to do with convexity. If the second derivative is positive, then the function is convex, which also means that all the tangent lines must stay below the graph of the function, at least near the point of tangency. Thus a maximum cannot occur where the derivative is positive.

Answer 4

It tells you whether it is a *local* max/min.

Your function may be bending at the max or min. If it's shaped like a cave (y=-x^2, say) then it is a max. If it's shaped like a bowl, it's a min.

The sign of the second derivative tells you which way the function is bending at the point. Negative sign means cave-like, so max. Positive means min.

As to why it works, the second derivative is the rate of change of the tangent line (the first derivative) of your function. A negative sign tells you that if you shift your tangent line over a tiny amount, it will be angled down a little. Your tangent line is like a sled starting to go down a hill. You must be at a max.

Answer 5

the second derivative talks about the tendency of a function to increase its slope or decrease its slope over the domain of the function. If we designate a point as a "local" minimum or maximum, then it can only be defined as a global minimum or maximum if we can show that the tendancy of the function is to continue along a path that does not increase or decrease (depending upon the point being a max or min) its slope such that the function value would become greater or less than the identified extrema.

Answer 6

The second derivitive measures rate of change in the first derivitive. This may sound obscure; in terms of the original function, a second derivitive=0 indicates that the TREND of the rate of change of the function which has been increasing or decreasing has reversed. Thus if you have local maximum/minimums, you would expect at least one second derivitive =0 point.