Advertisements
Advertisements
Question
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Advertisements
Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \left\{ x\left( x - 2 \right) \right\}^2 \]
\[ = \left( x^2 - 2x \right)^2 \]
\[ = x^4 + 4 x^2 - 4 x^3 \]
\[f'\left( x \right) = 4 x^3 + 8x - 12 x^2 \]
\[ = 4x \left( x^2 - 3x + 2 \right)\]
\[ = 4x \left( x - 1 \right)\left( x - 2 \right)\]
\[\text { Here, 0, 1 and 2 are the critical points}.\]
\[\text { The possible intervals are }\left( - \infty , 0 \right),\left( 0, 1 \right),\left( 1, 2 \right)\text { and }\left( 2, \infty \right).\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 4x \left( x - 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 2 \right) > 0\]
\[ \Rightarrow x \in \left( 0, 1 \right) \cup \left( 2, \infty \right) \]
\[\text { So,f(x)is increasing on x } \in \left( 0, 1 \right) \cup \left( 2, \infty \right) . \]
\[\text { For } f(x)\text { to be decreasing, we must have } \]
\[f'(x) < 0\]
\[ \Rightarrow 4x\left( x - 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x\left( x - 1 \right)\left( x - 2 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right)\]
\[\text { So, f(x) is decreasing on x } \in \left( - \infty , 0 \right) \cup \left( 1, 2 \right) .\]
APPEARS IN
RELATED QUESTIONS
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Show that f(x) = x – cos x is increasing for all x.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
y = log x satisfies for x > 1, the inequality ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?
If \[f'(x)<0\] throughout an interval, then \[f\] is
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
