Increasing and decreasing functions

In summary, the given function f(x)=x[ax-x^2]^(1/2) for a>0 is said to be increasing in (0,3a/4) and decreasing in (3a/4,a). Therefore, the correct option is C) both A,B. The provided answer of D) None of these is incorrect.
  • #1
Tanishq Nandan
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5

Homework Statement


f(x)=x[ax-x^2]^ (1/2) for a>0
Then,f(x)
A)increases on (3a/4 , a)
B)decreases on (0, 3a/4)
C)both A,B
D)None of these

Homework Equations


differentiation chain rule
f(x) is said to be increasing in (a,b) if it's derivative is positive and decreasing if it's derivative is negative for all x b/w a and b

The Attempt at a Solution


First af all,I found the domain of the given function which came out to be [0,a]
Now,the derivative of the function is:
(3ax-4x^2)/ [(4ax-4x^2)^1/2]
Now,the term in the denominator being inside square root is always positive,so we only need to deal with the numerator.
Which is:
x(3a-4x)
Now,due to it's domain x is also positive
Therefore the first term of the numerator is also positive,so it all comes down to the second term..
(3a-4x) which is positive(and hence the function increasing) for x b/w 0 and 3a/4 ,and negative for the rest.So,the corresponding option comes out to be D.
But,the answer given is C.
If anybody can point out where I am going wrong (or if the answer given is wrong,whichever),it qould be very helpful..
 
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  • #2
The answer given is wrong. Since the function is positive and goes to zero at x = 0 and x = a, it must increase in the beginning an decrease in the end. Here is a plot for a = 1:
upload_2017-7-1_22-22-13.png


Edit: If you don't want to bother differentiating the square root, you can also note that the function is positive in the domain and therefore is increasing/decreasing if its square is. It is much more convenient to differentiate ##ax^3 - x^4##.
 
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Likes Tanishq Nandan
  • #3
Got it
 

1. What is an increasing function?

An increasing function is a type of mathematical function where the output value increases as the input value increases. This means that as the input value increases, the output value also increases, resulting in a positive slope on a graph.

2. How can you identify an increasing function?

To identify an increasing function, you can look at the slope of the graph. If the slope is positive, then the function is increasing. Additionally, you can also look at the function's equation and determine if the coefficient of the independent variable is positive.

3. What is a decreasing function?

A decreasing function is a type of mathematical function where the output value decreases as the input value increases. This means that as the input value increases, the output value decreases, resulting in a negative slope on a graph.

4. How can you determine if a function is increasing or decreasing on a given interval?

To determine if a function is increasing or decreasing on a given interval, you can take the derivative of the function and evaluate it at different points within the interval. If the derivative is positive, then the function is increasing, and if the derivative is negative, then the function is decreasing.

5. Can a function be both increasing and decreasing?

No, a function cannot be both increasing and decreasing. A function can only be classified as either increasing or decreasing on a given interval. However, a function can have different intervals where it is increasing and other intervals where it is decreasing.

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