Question

The function \[f\left( x \right) = {\left( {9 - {x^2}} \right)^2}\] increases in
A. \[\left( { - 3,0} \right) \cup \left( {3,\infty } \right)\]
B. \[\left( { - \infty , - 3} \right) \cup \left( {3,\infty } \right)\]
C. \[\left( { - \infty , - 3} \right) \cup \left( {0,3} \right)\]
D. \[\left( { - 3,3} \right)\]
E. \[\left( {3,\infty } \right)\]

Answer 1

Hint: In this question, we are asked to find the range of the given function at which it is increasing. For that, we will use the concept of increasing function. First, we take the derivative of the function then simplify it and apply algebraic identity to get the desired result.

Formula used:
We have been using the following formulas:
1.\[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\]
2. \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\]

Complete step-by-step solution:
We are given a function \[f\left( x \right) = {\left( {9 - {x^2}} \right)^2}\]
Now we know that if the derivative of a function is greater than zero, it is an increasing function.
Now to find whether the given function is increasing or decreasing. For that, we take the derivative of the given function
\[f^{'}\left ( x \right )=2\left ( 9-x^{2} \right )\times\left ( -2x \right )\]
 Now by simplifying the above function, we get
\[
  f^{'}\left( x \right) = \left( { - 4x} \right)\left( {9 - {x^2}} \right) \\
   = \left( { - 4x} \right)\left( {{3^2} - {x^2}} \right) \]
Now we rearrange the above function as:
\[f^{'}\left( x \right) = \left( {4x} \right)\left( {{x^2} - {3^2}} \right)\]
We know that \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\]
By applying this algebraic identity in the above function, we get
\[f^{'}\left( x \right) = \left( {4x} \right)\left( {x - 3} \right)\left( {x + 3} \right)\]
Now, the question ask us to find the intervals in which given function is an increasing function. So we have to check for which interval of x derivative of f(x) > 0.
So, we need to find the critical points by equating the derivative of f(x) equal to 0.
So, \[f^{'}\left( x \right) = 0\]
Therefore, \[\,\,4x\left( {x - 3} \right)\left( {x + 3} \right) = 0\]
 When \[4x = 0\]
\[x = 0\]
When \[x - 3 = 0\]
\[x = 3\]
When \[x + 3 = 0\]
\[x = - 3\]
Thus, \[x = 0, \pm 3\]
Therefore, \[ - 3,0,3\] are the critical points.
The following table describes the nature of the given function.

Intervals using Critical Points	Nature of \[f{'}\left( x \right)\]	Nature of \[f\left( x \right)\]
\[\left( { - \infty , - 3} \right)\]	\[f{'}\left( x \right) < 0\]	Decreasing Function
\[\left( { - 3,0} \right)\]	\[f{'}\left( x \right) > 0\]	Increasing Function
\[\left( {0,3} \right)\]	\[f{'}\left( x \right) < 0\]	Decreasing Function
\[\left( {3,\infty } \right)\]	\[f{'}\left( x \right) > 0\]	Increasing Function

Therefore, \[f\left( x \right) = {\left( {9 - {x^2}} \right)^2}\] is increasing at \[\left( { - 3,0} \right) \cup \left( {3,\infty } \right)\]
Hence, option (A) is correct

Note: Students should be careful while taking the derivative of the given function and also be careful while applying the algebraic identity, as this mistake will affect our required result.