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MitsuShai
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Homework Statement
Given
Homework Equations
The Attempt at a Solution
Last edited:
rock.freak667 said:If I remember correctly, f(x) is an increasing function for f'(x) > 0 and a decreasing function for f'(x)< 0.
rock.freak667 said:How did you get that?
Char. Limit said:Now how about d and e?
Here's a clue...
What is the shape of the graph when the derivative is zero?
Char. Limit said:Hmm...
I'm sorry, but I don't get (-1,3) for increasing...
The derivative is this, as you have written, correct?
[tex]\frac{5(1-x)(x-1)}{(3x^2-10x+3)^2}[/tex]
Ignore the denominator... where is the numerator zero? Where are each of the three components zero?
Char. Limit said:I said ignore the denominator... if the denominator is zero, your graph is really screwing up.
OK, so you know the graph of the derivative is zero at -1 and 1.
Thus, the minimum and the maximum must be at those two points.
vela said:There's a singularity at x=1/3, which is in (-1,1). That's probably why.
Char. Limit said:Oh...
So it would be -1<x<1, x=/=(1/3) then...
vela said:Yes, exactly. The function isn't increasing at x=1/3 because it isn't defined there, so you have to split the interval as you have done.
vela said:What happens between x=-2 and x=-1?
Derivatives are a mathematical concept that represents the rate of change of a function at a specific point. In other words, it measures how much a function is changing at a particular input value.
The derivative of a function at a given point is equal to the slope of the tangent line at that point on the graph. This means that the derivative can tell us the direction and steepness of the slope at any point on the graph.
The average rate of change is the overall change in a function over a given interval, while the instantaneous rate of change is the change at a specific point. The derivative represents the instantaneous rate of change.
The chain rule is a rule in calculus that allows us to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.
Derivatives have many real-life applications, including in physics, economics, and engineering. For example, derivatives can be used to calculate the velocity and acceleration of an object, determine optimal production levels in business, or design efficient structures in engineering.