# Find the second derivative of F (x)

• ~Sam~
In summary, The second derivative of F(x) = integ[a,x] (t-x)2 f(t) d(t) can be found by using the Leibniz integral rule twice. The first term will be x^2, while the last term involves differentiating the integrand, resulting in the integral from a to x of t*dt. This gives a second derivative of F'(x) = 3*x^2/2-a^2/2.
~Sam~

## Homework Statement

Assuming sufficient differentiability, find second derivative of F(x) = integ[a,x] (t-x)2 f(t) d(t)

## Homework Equations

Probably Fund.Thm of Calculus and some properties

## The Attempt at a Solution

I really have no idea..I tried evaluating but with t=x but I get zero..I have never done second derivative so I'm somewhat clueless. Anyone care to help and explain?

The second derivative is the derivative of the first derivative. E.g., if f(x) = x^2, f'(x) = 2x, and f''(x) = d/dx(f'(x)) = d/dx(2x) = 2.

The Fundamental Theorem of Calculus can help you get F'(x), so take a careful look at this theorem and any examples that show how to use it. After that, just take the derivative of what you got for F'(x).

Use the Leibniz integral rule twice. Besides the term that you found equal to zero, there's another term that involves differentiating the integrand. That part is not necessarily zero.

I don't quite understand? I've never learned the leibniz rule. Can you exemplify it with this question? The f(t) part also confuses me.

Last edited:
Don't worry about the f(t) part. Your answer will be in terms of f(t). http://en.wikipedia.org/wiki/Leibniz_integral_rule Here's an easy example. Take F(x) to be the integral from a to x of x*t*dt. You only have to deal with the first and third terms in the rule. The first term you get just from putting t=x in the integrand, so x^2. That's the one you know. The last term says you should differentiate inside the integral. That gives you the integral from a to x of t*dt. Or x^2/2-t^2/2. Adding the two you get F'(x)=3*x^2/2-a^2/2. Compare that with what you get by actually integrating and then differentiating. Now try doing that with your example. What's F'(x)?

## 1. What is the second derivative of F (x)?

The second derivative of F (x) is the derivative of the derivative of F (x). It represents the rate of change of the slope of the original function F (x).

## 2. Why is it important to find the second derivative of F (x)?

Finding the second derivative of F (x) allows us to determine the concavity of the original function. This information can be used to analyze the behavior of the function and make predictions about its critical points, inflection points, and overall shape.

## 3. How do you find the second derivative of F (x)?

To find the second derivative of F (x), we first find the derivative of the original function F (x) using the rules of differentiation. Then, we take the derivative of the resulting function to get the second derivative. This can be done using the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the function.

## 4. Can the second derivative of F (x) be negative?

Yes, the second derivative of F (x) can be negative. This indicates that the original function is concave down, or "smiling," at that point. It also means that the rate of change of the slope of the function is decreasing at that point.

## 5. What does it mean if the second derivative of F (x) is equal to zero?

If the second derivative of F (x) is equal to zero, it means that the original function has an inflection point at that point. This means that the concavity of the function changes from concave up to concave down or vice versa. It also indicates that the rate of change of the slope of the function is neither increasing nor decreasing at that point.

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