Application of Differentiation Problem - Need help

In summary, the conversation is about a problem where the graph of a cubic polynomial is given, and the task is to prove the equation and find the coordinates of the local maximum using calculus. The given information includes the graph cutting through the origin, touching the x-axis at (3,0), and passing through (4,4). The suggested solution for part a) is to expand the given equation, while the difficulty for part b) is not specified.
  • #1
Lyle1
1
0
Hi,
I have a problem of which I do not know where to start or how to go about solving it.

- The graph of the function equation y=f(x) is shown, where f(x) is a cubic polynomial. The graph cuts through the origin, touches the x-axis at (3,0) and passes through (4,4).

a) - Prove that f(x) = x3-6x2+9x

b) - Use calculus to find the coordinates of the local maximum of the graphAny help would be appreciated. Thanks in advance (:
 
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  • #2
Lyle said:
Hi,
I have a problem of which I do not know where to start or how to go about solving it.

- The graph of the function equation y=f(x) is shown, where f(x) is a cubic polynomial. The graph cuts through the origin, touches the x-axis at (3,0) and passes through (4,4).

a) - Prove that f(x) = x3-6x2+9x

b) - Use calculus to find the coordinates of the local maximum of the graphAny help would be appreciated. Thanks in advance (:

You have three points, two zeros, and a y-intercept. Are you SURE you don't know where to start?
 
  • #3
a) As 0 is a zero of f(x) and there is a double root at 3, the equation of f(x) is x(x - 3)$^2$. When expanded this is equivalent to the equation you are given. Whether this is rigorous enough to be considered proof will be up to you.

What difficulty are you having with part b)?
 

1. What is differentiation and how is it used in problem-solving?

Differentiation is a mathematical tool used to find the rate of change of a function. It is particularly useful in problem-solving as it allows us to analyze the behavior of a function and make predictions about its future values.

2. What are some common real-world applications of differentiation?

Differentiation is used in various fields such as physics, engineering, economics, and biology. Some common real-world applications include determining the velocity and acceleration of an object, finding the maximum or minimum points of a function, and analyzing the growth rate of a population.

3. Can you provide an example of a differentiation problem and its solution?

Sure, let's say we have a function f(x) = x^2. To find the slope of the tangent line at a specific point, say x = 3, we can use the derivative of the function, which is f'(x) = 2x. Plugging in x = 3, we get f'(3) = 2(3) = 6. Therefore, the slope of the tangent line at x = 3 is 6.

4. How does the chain rule work in differentiation?

The chain rule is a rule used to find the derivative of a composite function. It states that the derivative of a composite function is equal to the product of the derivatives of its individual functions. In other words, if we have a function f(g(x)), the chain rule states that f'(g(x)) * g'(x) = f'(x).

5. What are some tips for solving differentiation problems?

Here are some tips for solving differentiation problems: 1) Familiarize yourself with the basic rules of differentiation, such as the power rule, product rule, and chain rule. 2) Practice solving problems to improve your understanding and speed. 3) Pay attention to the given function and carefully choose the appropriate rule to use. 4) Double-check your solution for accuracy. 5) Seek help from peers or a tutor if needed.

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