9000070302
Level:
B
Given function \(f(x) = x^{3} + 3x^{2} + 12x + 4\), find
the intervals where \(f\)
is a strictly concave down function.
\((-\infty ,-1)\)
\((-\infty ,0)\)
\((-\infty ,2)\)
\((-\infty ,4)\)
B
9000070702
Level:
B
Differentiate the following function.
\[
f(x) = (x^{2} - 3x + 2)^{\frac{1}
{2} }
\]
\(f^{\prime}(x) = \frac{2x-3}
{2\sqrt{x^{2 } -3x+2}},\ x\in \mathbb{R}\setminus \left [ 1,2\right ] \)
\(f^{\prime}(x) = \frac{2x-3}
{2\sqrt{x^{2 } -3x+2}},\ x\in \mathbb{R}\setminus \left (1,2\right )\)
\(f^{\prime}(x) = (4x - 6)\sqrt{x^{2 } - 3x + 2},\ x\in \mathbb{R}\setminus \left [ 1,2\right ] \)
\(f^{\prime}(x) = (4x - 6)\sqrt{x^{2 } - 3x + 2},\ x\in \mathbb{R}\setminus \left (1,2\right )\)
9000070303
Level:
B
Given function \(f(x) = -x^{3} + 6x^{2} + 6x + 1\), find
the intervals where \(f\)
is a strictly concave up function.
\((-\infty ,2)\)
\((-1,\infty )\)
\((-\infty ,3)\)
\((-\infty ,6)\)
9000070703
Level:
B
Differentiate the following function.
\[
f(x)= \sqrt{\sin x -\cos x}
\]
\(f^{\prime}(x) = \frac{\sin x+\cos x}
{2\sqrt{\sin x-\cos x}},\ x\in \left ( \frac{\pi }{4} + 2k\pi , \frac{5\pi }
{4} + 2k\pi \right ),\ k\in \mathbb{Z}\)
\(f^{\prime}(x) = \frac{\sin x+\cos x}
{2\sqrt{\sin x-\cos x}},\ x\in \left [ \frac{\pi }{4} + 2k\pi , \frac{5\pi }
{4} + 2k\pi \right ] ,\ k\in \mathbb{Z}\)
\(f^{\prime}(x) = \frac{\sin x-\cos x}
{2\sqrt{\sin x-\cos x}},\ x\in \left [ \frac{\pi }{4} + 2k\pi , \frac{5\pi }
{4} + 2k\pi \right ] ,\ k\in \mathbb{Z}\)
\(f^{\prime}(x) = \frac{\sin x-\cos x}
{2\sqrt{\sin x-\cos x}},\ x\in \left ( \frac{\pi }{4} + 2k\pi , \frac{5\pi }
{4} + 2k\pi \right ),\ k\in \mathbb{Z}\)
9000070304
Level:
B
Given function \(f(x) = -x^{3} - 12x^{2} + 12x - 2\), find
the intervals where \(f\)
is a strictly concave up function.
\((-\infty ,-4)\)
\((-\infty ,2)\)
\((-\infty ,6)\)
\((-\infty ,12)\)
9000070704
Level:
B
Differentiate the following function.
\[
f(x) = \frac{1}
{\cos x + 3x^{2}}
\]
\(f^{\prime}(x) = \frac{\sin x-6x}
{(3x^{2}+\cos x)^{2}} ,\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x}
{(3x^{2}+\cos x)^{2}} ,\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{\sin x-6x}
{3x^{2}+\cos x},\ x\in \mathbb{R}\)
\(f^{\prime}(x) = \frac{6x-\sin x}
{3x^{2}+\cos x},\ x\in \mathbb{R}\)
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.
\((-3,2)\)
\((-3,4)\)
\((-4,2)\)
\((-2,3)\)
9000070705
Level:
B
Differentiate the following function.
\[
f(x) =\ln (2x^{2} + 5x)
\]
\(f^{\prime}(x) = \frac{4x+5}
{2x^{2}+5x},\ x\in \left (-\infty ,-\frac{5}
{2}\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{4x+5}
{2x^{2}+5x},\ x\in \mathbb{R}\setminus \left \{-\frac{5}
{2},0\right \}\)
\(f^{\prime}(x) = \frac{1}
{2x^{2}+5x},\ x\in \left (-\infty ,-\frac{5}
{2}\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{1}
{2x^{2}+5x},\ x\in \mathbb{R}\setminus \left \{-\frac{5}
{2},0\right \}\)
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.
\((-4,1)\)
\((-6,2)\)
\((2,4)\)
\((-5,4)\)
9000070706
Level:
B
Differentiate the following function.
\[
f(x) = \sqrt{x^{2 } + 3x}
\]
\(f^{\prime}(x) = \frac{2x+3}
{2\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3}
{2\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right ] \cup \left [ 0,\infty \right )\)
\(f^{\prime}(x) = \frac{2x+3}
{\sqrt{x^{2 } +3x}},\ x\in \left (-\infty ,-3\right )\cup \left (0,\infty \right )\)
\(f^{\prime}(x) = \frac{\sqrt{x^{2 } +3x}}
{2x+3} ,\ x\in \left (-\infty ,-3\right ] \cup \left [ 0,\infty \right )\)