Advertisements
Advertisements
Question
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Advertisements
Solution 1
Given:\[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\]
Differentiating w.r.t. x, we get:
f'(x) = \[6 x^3 - 12 x^2 - 90x\]
\[6x\left( x^2 - 2x - 15 \right)\] At critical points, f'(x)=0.
\[6x\left( x^2 - 2x - 15 \right)\] =0
\[\Rightarrow 6x\left( x^2 - 5x + 3x - 15 \right) = 0\]
\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) = 0\]
\[ \Rightarrow x = - 3, 0, 5\]
Solution 2
Given:\[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\]
Differentiating w.r.t. x, we get:
f'(x) = \[6 x^3 - 12 x^2 - 90x\]
\[6x\left( x^2 - 2x - 15 \right)\] At critical points, f'(x)=0.
\[6x\left( x^2 - 2x - 15 \right)\] =0
\[\Rightarrow 6x\left( x^2 - 5x + 3x - 15 \right) = 0\]
\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) = 0\]
\[ \Rightarrow x = - 3, 0, 5\]
| Interval | f'(x)= \[6x\left( x - 5 \right)\left( x + 3 \right)\] | Result |
| \[\left( - \infty , - 3 \right)\] | f'(-4)=-216 <0 | strictly decreasing |
| \[\left( - 3, 0 \right)\] | f'(-1)= 72 >0 | strictly increasing |
| \[\left( 0, 5 \right)\] | f'(1)= -96 <0 | strictly decreasing |
| \[\left( 5, \infty \right)\] | f'(6)=324 >0 | strictly increasing
|
(a) Hence the function is strictly increasing in \[\left( - 3, 0 \right)\] \[\cup\] \[\left( 5, \infty \right)\] .
(b) Also, the function is strictly decreasing in \[\left( - \infty , - 3 \right)\] \[\cup\] \[\left( 0, 5 \right)\] .
Solution 3
Given:\[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\]
Differentiating w.r.t. x, we get:
f'(x) = \[6 x^3 - 12 x^2 - 90x\]
\[6x\left( x^2 - 2x - 15 \right)\] At critical points, f'(x)=0.
\[6x\left( x^2 - 2x - 15 \right)\] =0
\[\Rightarrow 6x\left( x^2 - 5x + 3x - 15 \right) = 0\]
\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) = 0\]
\[ \Rightarrow x = - 3, 0, 5\]
| Interval | f'(x)= \[6x\left( x - 5 \right)\left( x + 3 \right)\] | Result |
| \[\left( - \infty , - 3 \right)\] | f'(-4)=-216 <0 | strictly decreasing |
| \[\left( - 3, 0 \right)\] | f'(-1)= 72 >0 | strictly increasing |
| \[\left( 0, 5 \right)\] | f'(1)= -96 <0 | strictly decreasing |
| \[\left( 5, \infty \right)\] | f'(6)=324 >0 | strictly increasing
|
(a) Hence the function is strictly increasing in \[\left( - 3, 0 \right)\] \[\cup\] \[\left( 5, \infty \right)\] .
(b) Also, the function is strictly decreasing in \[\left( - \infty , - 3 \right)\] \[\cup\] \[\left( 0, 5 \right)\] .
APPEARS IN
RELATED QUESTIONS
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
The function f(x) = cot−1 x + x increases in the interval
Let f(x) = x3 − 6x2 + 15x + 3. Then,
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f (x) = x2, for all real x, is ____________.
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
The function f(x) = x3 + 3x is increasing in interval ______.
