What is the proof for f(1.1)>-0.1?

In summary, a differentiable function is a mathematical function that is continuous and has a well-defined derivative at every point in its domain. This means that the function has a smooth and continuous graph with no sharp corners or breaks, and the rate of change of the function at any point can be calculated. A differentiable function is different from a non-differentiable function in that it has a smoother and more predictable behavior. Differentiable functions are important in mathematics for modeling and analyzing real-world phenomena, and their derivatives can be calculated using various rules. Not all functions can be differentiated, as they must be continuous and have a well-defined derivative at every point in their domain.
  • #1
Yankel
395
0
Hello all,

I am not sure how to approach this question:

Let f(x) be a continuous and differentiable function of order 2. Let f''(x) >0 for all values of x. The tangent line to the function at x=1 is y=-x+1. Show that f(1.1)>-0.1.

Thanks!
 
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  • #2
I would begin with:

\(\displaystyle f(x)=ax^2+bx+c\)

You are given:

\(\displaystyle a>0\)

\(\displaystyle f(1)=0\)

\(\displaystyle f'(1)=-1\)

Use these data to write $b$ and $c$ in terms of $a$. You should then be able to demonstrate that:

\(\displaystyle f(1.1)=0.01a-0.1>-0.1\)
 

FAQ: What is the proof for f(1.1)>-0.1?

What is a differentiable function?

A differentiable function is a mathematical function that is continuous and has a well-defined derivative at every point in its domain. This means that the function has a smooth and continuous graph with no sharp corners or breaks, and the rate of change of the function at any point can be calculated.

How is a differentiable function different from a non-differentiable function?

A differentiable function is continuous and has a well-defined derivative at every point in its domain, while a non-differentiable function may have sharp corners, breaks, or other discontinuities in its graph. This means that a differentiable function has a smoother and more predictable behavior than a non-differentiable function.

What is the importance of differentiable functions in mathematics?

Differentiable functions are important in mathematics because they allow us to model and analyze real-world phenomena with precise and accurate mathematical tools. They also have many applications in physics, engineering, economics, and other fields.

How do you determine if a function is differentiable?

A function is differentiable if it is continuous and has a well-defined derivative at every point in its domain. The derivative of a function can be calculated using the limit definition or through various differentiation rules, such as the power rule, product rule, and chain rule.

Can all functions be differentiated?

No, not all functions can be differentiated. For a function to be differentiable, it must be continuous and have a well-defined derivative at every point in its domain. Functions that have sharp corners, breaks, or other discontinuities in their graphs are not differentiable. Additionally, some functions may not have a well-defined derivative at certain points, such as at points where the function is not continuous.

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