Let n ≥ 2 be a natural number and f:[0,1)]\(\rightarrow\) R be the function defined by
If n is such that the area of the region bounded by the curves x = 0, x = 1, y = 0 and y = f(x) is 4, then the maximum value of the function f is
If n is such that the area of the region bounded by the curves x = 0, x = 1, y = 0 and y = f(x) is 4, then the maximum value of the function f is
Correct Answer: 8
Approach Solution - 1
Analysis of the Piecewise Function
Given: The piecewise function is defined as:
\(f(x) = \begin{cases} n(1 - 2nx), & \text{if } 0 \leq x < \frac{1}{2n} \\ 2n(2n - 1), & \text{if } \frac{1}{2n} \leq x < \frac{3}{4n} \\ 4n(1 - nx), & \text{if } \frac{3}{4n} \leq x < \frac{1}{n} \\ \frac{n}{n - 1}(nx - 1), & \text{if } \frac{1}{n} \leq x \leq 1 \end{cases}\)
The function \( f(x) \) is defined over the interval \( x \in [0, 1] \).
Solution:
We need to determine the intervals where \( f(x) \) is increasing or decreasing.
We can calculate the derivative of \( f(x) \) for each piece and find the intervals where the derivative is positive (increasing) or negative (decreasing).
By examining the given intervals and the derivatives, we can summarize the behavior of \( f(x) \):
- \( f(x) \) is decreasing in \( \left[0, \frac{1}{2n}\right) \)
- \( f(x) \) is increasing in \( \left[\frac{1}{2n}, \frac{3}{4n}\right) \)
- \( f(x) \) is decreasing in \( \left[\frac{3}{4n}, \frac{1}{n}\right) \)
- \( f(x) \) is increasing in \( \left[\frac{1}{n}, 1\right] \)
Conclusion:
The correct answer is 8.
Approach Solution -2
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C