Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\]
\[f'\left( x \right) = 6 x^3 - 12 x^2 - 90x\]
\[ = 6x\left( x^2 - 2x - 15 \right)\]
\[ = 6x\left( x - 5 \right)\left( x + 3 \right)\]
\[\text { Here, } x = - 3, x = 0 \text { and }x = 5 \text { are the critical points }.\]
\[\text { The possible intervals are }\left( - \infty , - 3 \right),\left( - 3, 0 \right),\left( 0, 5 \right)\text { and }\left( 5, \infty \right). .....(1)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) > 0 \left[\text { Since,} 6 > 0, 6x\left( x - 5 \right)\left( x + 3 \right) > 0 \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) > 0 \right]\]
\[ \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) > 0\]
\[ \Rightarrow x \in \left( - 3, 0 \right) \cup \left( 5, \infty \right) \left[ \text { From eq.} (1) \right]\]
\[\text { So,f(x)is increasing on x } \in \left( - 3, 0 \right) \cup \left( 5, \infty \right) .\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) < 0 \left[ \text { Since }6 > 0, 6x\left( x - 5 \right)\left( x + 3 \right) < 0 \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) < 0 \right]\]
\[ \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty , - 3 \right) \cup \left( 0, 5 \right) \left[ \text { From eq.} (1) \right]\]
\[\text { So,f(x)is decreasing on x } \in \left( - \infty , - 3 \right) \cup \left( 0, 5 \right) .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on 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) = 2x3 + 9x2 + 12x + 20 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Which of the following functions is decreasing on `(0, pi/2)`?
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
A function f is said to be increasing at a point c if ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
