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Yes.

Q: If derivative of f equals 0 for all x in the domain of f then is f a constant function?

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The "double prime", or second derivative of y = 5x, equals zero. The first derivative is 5, a constant. Since the derivative of any constant is zero, the derivative of 5 is zero.

Give the domain for

f(x)=1 f'(x)=0 because the derivative of a constant is ALWAYS 0.

Yes, but only if the domain is the real numbers. The derivative is y = 1.

pi divided by 6 is a constant and so its first derivative is 0. And since that is also a constant, the second derivative is 0. It is not clear what f(x) = csc(x) has to do with that!

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The "double prime", or second derivative of y = 5x, equals zero. The first derivative is 5, a constant. Since the derivative of any constant is zero, the derivative of 5 is zero.

you have to first find the derivative of the original function. You then make the derivative equal to zero and solve for x.

Give the domain for

Because the derivative of e^x is e^x (the original function back again). This is the only function that has this behavior.

f(x)=1 f'(x)=0 because the derivative of a constant is ALWAYS 0.

Yes, but only if the domain is the real numbers. The derivative is y = 1.

The highest point on a graph is when the derivative of the graph equals 0 or the slope is constant.

13

pi divided by 6 is a constant and so its first derivative is 0. And since that is also a constant, the second derivative is 0. It is not clear what f(x) = csc(x) has to do with that!

y is a sum of constants and so is itself a constant. Its derivative is, therefore, zero.

The "critical points" of a function are the points at which the derivative equals zero or the derivative is undefined. To find the critical points, you first find the derivative of the function. You then set that derivative equal to zero. Any values at which the derivative equals zero are "critical points". You then determine if the derivative is ever undefined at a point (for example, because the denominator of a fraction is equal to zero at that point). Any such points are also called "critical points". In essence, the critical points are the relative minima or maxima of a function (where the graph of the function reverses direction) and can be easily determined by visually examining the graph.

The range depends on the domain, which is not specified.