Advertisements
Advertisements
प्रश्न
What are the values of 'a' for which f(x) = ax is increasing on R ?
Advertisements
उत्तर
\[f\left( x \right) = a^x \]
\[f'\left( x \right) = a^x \log a\]
\[\text { Given }: f(x) \text { is increasing on R } . \]
\[ \Rightarrow f'\left( x \right) > 0\]
\[ \Rightarrow a^x \log a > 0\]
\[\text { Logarithmic function is defined for positive values of a } . \]
\[ \Rightarrow a > 0\]
\[ \Rightarrow a^x > 0\]
\[\text { We know,} \]
\[ a^x \log a > 0\]
\[\text{ It can be possible when } a^x > 0 \text { and } \log a > 0 \text { or }a^x < 0 \text { and } \log a < 0 . \]
\[ \Rightarrow \log a > 0\]
\[ \Rightarrow a > 1\]
\[\text { So, }f(x)\text { is increasing when }a> 1 .\]
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is
- Strictly increasing
- Strictly decreasing
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Show that f(x) = e2x is increasing on R.
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) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that f(x) = x – cos x is increasing for all x.
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
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`
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The function f(x) = tanx – x ______.
The function f (x) = x2, for all real x, is ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
y = log x satisfies for x > 1, the inequality ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
