Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 2 x^3 + 9 x^2 + 12x + 20\]
\[f'\left( x \right) = 6 x^2 + 18x + 12\]
\[ = 6 \left( x^2 + 3x + 2 \right)\]
\[ = 6 \left( x + 1 \right)\left( x + 2 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x + 1 \right)\left( x + 2 \right) > 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x + 2 \right) > 0 \left[ \text { Since } 6 > 0, 6 \left( x + 1 \right)\left( x + 2 \right) > 0 \Rightarrow \left( x + 1 \right)\left( x + 2 \right) > 0 \right]\]
\[ \Rightarrow x < - 2 \ or \ x > - 1\]
\[ \Rightarrow x \in \left( - \infty , - 2 \right) \cup \left( - 1, \infty \right)\]
\[\text { So },f(x)\text { is increasing on } x \in \left( - \infty , - 2 \right) \cup \left( - 1, \infty \right).\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x + 1 \right)\left( x + 2 \right) < 0\]
\[ \Rightarrow \left( x + 1 \right)\left( x + 2 \right) < 0 \left[ \text { Since } 6 > 0, 6 \left( x + 1 \right)\left( x + 2 \right) < 0 \Rightarrow \left( x + 1 \right)\left( x + 2 \right) < 0 \right]\]
\[ \Rightarrow - 2 < x < - 1 \]
\[ \Rightarrow x \in \left( - 2, - 1 \right)\]
\[\text { So, }f(x)\text { is decreasing on }x \in \left( - 2, - 1 \right) .\]
APPEARS IN
संबंधित प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
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 function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Show that f(x) = loga x, 0 < a < 1 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) ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
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) ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
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) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Every invertible function is
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = tanx – x ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f (x) = 2 – 3 x is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
