Solving for m and b in a differentiable function at x=2.

In summary, a function is said to be differentiable everywhere if it has a derivative at every point in its domain. To determine if a function is differentiable everywhere, you can use the definition of differentiability or check if the function is continuous and has a derivative at every point. The implications of a function being differentiable everywhere include it being smooth and allowing for the use of calculus techniques. Functions that are differentiable everywhere have real-world applications in various fields. A function cannot be differentiable everywhere if it is not continuous, as the definition of differentiability requires the function to be continuous at the point of differentiation.
  • #1
Rasine
208
0
f(x)= x^4 x less than or = 2
mx+b x is greater than 2

Find the values of m and b that make f differentiable everywhere.

so what i was trying to do was to find where the graph of x^4=mx where x=2 so that the mx+b function would start at where ever x^4 left off at x=2

i got m=8 and b=0...but that is not right...please tell me what i am doing wrong
 
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  • #2
At x=2:
x4=16
4x3=32

Therefore:
2m+b=16
m=32
resulting in b=-48.
 
  • #3
thank you so much!
 

1. What does it mean for a function to be differentiable everywhere?

A function is said to be differentiable everywhere if it has a derivative at every point in its domain. This means that the function is smooth and has a well-defined tangent line at every point.

2. How can I determine if a function is differentiable everywhere?

To determine if a function is differentiable everywhere, you can use the definition of differentiability, which states that a function is differentiable at a point if the limit of its difference quotient exists. Alternatively, you can also check if the function is continuous and has a derivative at every point in its domain.

3. What are the implications of a function being differentiable everywhere?

A function being differentiable everywhere implies that it is a smooth and well-behaved function. This means that it is easy to work with and allows for the use of calculus techniques, such as finding critical points and optimizing the function.

4. Are there any real-world applications of functions that are differentiable everywhere?

Functions that are differentiable everywhere have many real-world applications, especially in fields such as physics, engineering, and economics. For example, in physics, differentiable functions are used to model the motion of objects, while in economics, they are used to represent demand and supply curves.

5. Can a function be differentiable everywhere but not continuous?

No, a function cannot be differentiable everywhere if it is not continuous. This is because the definition of differentiability requires the function to be continuous at the point of differentiation. Therefore, if a function is not continuous, it cannot have a derivative at that point, and thus cannot be differentiable there.

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