Q: Does a differentiable function have to have a relative minimum between any two relative maximum?

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Differentiation lets you find the rate of change of a function. You can use this to find the maximum or minimum values of a differentiable function, which is useful in a lot of optimization problems. It's also necessary for differential equations, which are useful just about everywhere.

Yes.If you find 2 relative minima and the function is continuous, there must be exactly one point between these minima with the highest value in that interval. This point is a relative maxima.Think of temperature for example (it is continuous).

The range is a measure of the difference between the maximum and minimum values that a variable can take, or that a function can take over the relevant domain.

No, since the equation could be y = x3 (or something similar) which will have a point of inflection at (0,0), meaning there is no relative maximum/minimum, as the graph doesn't double back on itself For those that are unfamiliar with a point of inflection <http://mathsfirst.massey.ac.nz/Calculus/SignsOfDer/images/Introduction/POIinc.png>

The "maximum" function.

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Differentiation lets you find the rate of change of a function. You can use this to find the maximum or minimum values of a differentiable function, which is useful in a lot of optimization problems. It's also necessary for differential equations, which are useful just about everywhere.

the first or the last term of a proportion or series. a relative maximum or relative minimum value of a function in a given region.

It depends on the function whose maximum you are trying to find. If it is a well behaved function over the domain in question, you could differentiate it and set its derivative equal to 0. Solve the resulting equation for possible stationary points. Evaluate the second derivative at these points and, if that is negative, you have a maximum. If the second derivative is also 0, then you have to go to higher derivatives (if they exist). If the function is not differentiable, you may have a more difficult task at hand.

Yes.If you find 2 relative minima and the function is continuous, there must be exactly one point between these minima with the highest value in that interval. This point is a relative maxima.Think of temperature for example (it is continuous).

It depends on the function and the level of your mathematical skills. It also depends on whether you are looking for a global maximum or a local one. For example, a cubic equation [y = ax3 + bx2 + cd + d] has no global maximum but it will usually have a local one. This is also the case for any equation that is asymptotically +infinity somewhere in its domain. If a function is twice differentiable over the domain in question, differentiate it once, set the resulting derivative equal to zero and solve for the coordinates of the stationary point. Next, differentiate it again and evaluate the value of the second derivative at the stationary point. If this derivative is negative, you have a local maximum at the stationary point. But be careful at the edges of the domain. All this does not help if the function is not twice differentiable. Sometimes there are other ways. For example, let P(X = x) be the [probability distribution] function that X, the sum of the numbers on two dice, is x. Then it can easily be shown that P(2) = P(12) = 1/36 P(3) = P(11) = 2/36 P(4) = P(10) = 3/36 P(5) = P(9) = 4/36 P(6) = P(8) = 5/36 P(7) = = 6/36 and P(x) = 0 elsewhere. P is not a continuous function and so cannot be differentiable, but the table above shows that the maximum of the function is at P(7).

It depends on the function. Some functions, for example any polynomial of odd order, will have no maximum or minimum. Some functions, such as the sine or cosine functions, will have an infinite number of maxima and minima. If a function is differentiable then a turning point can be found by finding the zero of its derivative. This could be a maximum, minimum or a point of inflexion. If the derivative before this zero is negative and after the zero is positive then the point is a minimum. If it goes from positive to negative, the pont is a maximum, and if it has the same sign (either both +ve or both -ve) then it is a point of inflexion. A second derivative can help answer this quicker, but it need not exist. These are all well behaved functions. The task is much more complicated for ill behaved functions. Consider, for example, the difference between consecutive primes. The minimum is clearly 1 (between 2 and 3) but the maximum? Or the number of digits between 1 and 4 in the decimal expansio of pi = 3.14159.... Minimum digit between = 0 (they are consecutive near the start of pi), but maximum?

You take the derivative of the function. The derivative is another function that tells you the slope of the original function at any point. (If you don't know about derivatives already, you can learn the details on how to calculate in a calculus textbook. Or read the Wikipedia article for a brief introduction.) Once you have the derivative, you solve it for zero (derivative = 0). Any local maximum or minimum either has a derivative of zero, has no defined derivative, or is a border point (on the border of the interval you are considering). Now, as to the intervals where the function increase or decreases: Between any such maximum or minimum points, you take any random point and check whether the derivative is positive or negative. If it is positive, the function is increasing.

Addition is the maximum or minimum function in math.

The range is a measure of the difference between the maximum and minimum values that a variable can take, or that a function can take over the relevant domain.

The maximum value a wave reaches relative to its resting position is called the amplitude. It represents the maximum displacement of the wave from its equilibrium position.

By taking the derivative of the function. At the maximum or minimum of a function, the derivative is zero, or doesn't exist. And end-point of the domain where the function is defined may also be a maximum or minimum.

No, since the equation could be y = x3 (or something similar) which will have a point of inflection at (0,0), meaning there is no relative maximum/minimum, as the graph doesn't double back on itself For those that are unfamiliar with a point of inflection <http://mathsfirst.massey.ac.nz/Calculus/SignsOfDer/images/Introduction/POIinc.png>