Advertisements
Advertisements
प्रश्न
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Advertisements
उत्तर
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = x^3 - 6 x^2 + 9x + 15\]
\[f'\left( x \right) = 3 x^2 - 12x + 9\]
\[ = 3 \left( x^2 - 4x + 3 \right)\]
\[ = 3 \left( x - 1 \right)\left( x - 3 \right)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 3 \left( x - 1 \right)\left( x - 3 \right) > 0 \]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \left[ \text { Since } 3 > 0, 3 \left( x - 1 \right)\left( x - 3 \right) > 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \right]\]
\[ \Rightarrow x < 1 \ or \ x > 3\]
\[ \Rightarrow x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right)\]
\[\text { So,f(x)is increasing on } x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right).\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 3 \left( x - 1 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \left[ \text { Since } 3 > 0, 3 \left( x - 1 \right)\left( x - 3 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \right]\]
\[ \Rightarrow 1 < x < 3 \]
\[ \Rightarrow x \in \left( 1, 3 \right)\]
\[\text { So,f(x)is decreasing on x } \in \left( 1, 3 \right) .\]
APPEARS IN
संबंधित प्रश्न
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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 \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
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] ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
The function f(x) = xx decreases on the interval
The function f(x) = x2 e−x is monotonic increasing when
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
The function f(x) = x9 + 3x7 + 64 is increasing on
Find `dy/dx,if e^x+e^y=e^(x-y)`
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
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] ______
The function f(x) = sin x + 2x is ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The function f(x) = tanx – x ______.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function `"f"("x") = "x"/"logx"` increases on the interval
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
Which of the following graph represent the strictly increasing function.
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.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
