9000070304
Level:
B
Given function \(f(x) = -x^{3} - 12x^{2} + 12x - 2\), find
the intervals where \(f\)
is a strictly concave up function.
\((-\infty ;-4)\)
\((-\infty ;2)\)
\((-\infty ;6)\)
\((-\infty ;12)\)
B
9000070707
Level:
B
Differentiate the following function.
\[
f(x) = \root{5}\of{x^{2} - 7x}
\]Remark: The function \(f\colon y = \root{5}\of{x}\)
is defined for \(x\in \left < 0;\infty \right )\).
\(f^{\prime}(x) = \frac{2x-7}
{5(x^{2}-7x)^{\frac{4}
{5} }} ;\ x\in \left (-\infty ;0\right )\cup \left (7;\infty \right )\)
\(f^{\prime}(x) = \frac{2x-7}
{5(x^{2}-7x)^{\frac{4}
{5} }} ;\ x\in \left (-\infty ;0\right ] \cup \left [ 7;\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x};\ x\in \left (-\infty ;0\right )\cup \left (7;\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x};\ x\in \left (-\infty ;0\right ] \cup \left [ 7;\infty \right )\)
9000070305
Level:
B
Given function \(f(x) = x^{4} + 2x^{3} - 36x^{2} + 36x + 2\), find
the intervals where \(f\)
is a strictly concave down function.
\((-3;2)\)
\((-3;4)\)
\((-4;2)\)
\((-2;3)\)
9000070801
Level:
B
Differentiate the following function.
\[
f(x) = 3\sin x\cos x
\]
\(f'(x) = 3\cos (2x);\ x\in \mathbb{R}\)
\(f'(x) = 3;\ x\in \mathbb{R}\)
\(f'(x) = -3\cos x\sin x;\ x\in \mathbb{R}\)
\(f'(x) = 3(\cos x)^{2};\ x\in \mathbb{R}\)
9000070306
Level:
B
Given function \(f(x) = x^{4} + 6x^{3} - 24x^{2} + x + 3\), find
the intervals where \(f\)
is a strictly concave down function.
\((-4;1)\)
\((-6;2)\)
\((2;4)\)
\((-5;4)\)
9000070807
Level:
B
Differentiate the following function.
\[
f(x) = \frac{x^{4} + 3}
{x^{2}} + x^{3}
\]
\(f'(x) = 3x^{2} + 2x - \frac{6}
{x^{3}} ;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 6x^{2} - 2x - \frac{6}
{x^{3}} ;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 3x^{2} + 2x + \frac{6}
{x^{3}} ;\ x\in \mathbb{R}\setminus \{0\}\)
\(f'(x) = 6x^{2} - 2x + \frac{6}
{x^{3}} ;\ x\in \mathbb{R}\setminus \{0\}\)
9000070501
Level:
B
Consider the geometric sequence with \(a_{2} = 50\)
and \(a_{3} = 25\).
Find the sum of the first four terms.
\(187.5\)
\(93.75\)
\(250\)
\(375\)
\(500\)
9000070808
Level:
B
Differentiate the following function.
\[
f(x)= \frac{x}
{x + 1}
\]
\(f'(x) = \frac{1}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{1}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = \frac{x}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
\(f'(x) = - \frac{x}
{(x+1)^{2}} ;\ x\in \mathbb{R}\setminus \{ - 1\}\)
9000070809
Level:
B
Differentiate the following function.
\[
f(x)= 3x^{2}\sin x
\]
\(f'(x) = 6x\sin x + 3x^{2}\cos x;\ x\in \mathbb{R}\)
\(f'(x) = 6x\cos x;\ x\in \mathbb{R}\)
\(f'(x) = 3x^{2}\sin x\cos x;\ x\in \mathbb{R}\)
\(f'(x) = -3x^{2}\sin x\cos x;\ x\in \mathbb{R}\)
9000070502
Level:
B
Consider the geometric sequence with the first term
\(a_{1} = 243\) and quotient
\(q = \frac{1}
{3}\). Find
\(n\) such that the
sum of the first \(n\)
terms equals \(363\).
\(n = 5\)
\(n = 2\)
\(n = 3\)
\(n = 4\)
\(n = 6\)