Question
Prove that the following function are increasing on R.
$f(x) = 3x^5 + 40x^3 + 240x$
$f(x) = 3x^5 + 40x^3 + 240x$
✓
Answer
$f(x) = 3x^5 + 40x^3 + 240x$
$f'(x) = 15x^4 + 120x^2 + 240$
$= 15(x^4 + 8x^2 +16)$
$=15(\text{x}^2+4)^2>0,\forall\ \text{x}\in\text{R}$ $[\because\ 15>0\text{ and }(\text{x}^2+4)^2>0]$
So, f(x) increasing on R.
$f'(x) = 15x^4 + 120x^2 + 240$
$= 15(x^4 + 8x^2 +16)$
$=15(\text{x}^2+4)^2>0,\forall\ \text{x}\in\text{R}$ $[\because\ 15>0\text{ and }(\text{x}^2+4)^2>0]$
So, f(x) increasing on R.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In the following matrix equation use elementary operation $R_2 → R_2 + R_1$ and the equation thus obtained:
$\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix}\begin{bmatrix} 8 & -3 \\ 9 & -4 \end{bmatrix}$
→$\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix}\begin{bmatrix} 8 & -3 \\ 9 & -4 \end{bmatrix}$
Compute $\left[\begin{array}{cc} {\cos ^{2} x} & {\sin ^{2} x} \\ {\sin ^{2} x} & {\cos ^{2} x} \end{array}\right]+\left[\begin{array}{cc} {\sin ^{2} x} & {\cos ^{2} x} \\ {\cos ^{2} x} & {\sin ^{2} x} \end{array}\right]$
→For a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by $\text{a}_{\text{ij}}=\frac{\text{i}}{\text{j}}$
, write the value of $a_{12}$.
→, write the value of $a_{12}$.If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are three mutually perpendicular unit vectors, then prove that $\big|\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big|=\sqrt{3}$
→Prove that the function does not have maxima or minima: g(x) = log x
Construct a 3$\times$4 matrix, whose elements are given by aij $ = \frac{1}{2}\left| { - 3i + j} \right|$
→For any two vectors $\vec{\text{a}} $ and $\vec{\text{b}},$ write when $\big|\vec{\text{a}}+\vec{\text{b}}\big|=|\vec{\text{a}}|+\big|\vec{\text{b}}\big|$ holds.
→If $2\begin{bmatrix}3&4\\5&\text{x} \end{bmatrix}+\begin{bmatrix}1&\text{y}\\0&1 \end{bmatrix}=\begin{bmatrix}7&0\\10&5 \end{bmatrix},$ find x - y.
→Find the value of $\theta\in(0,\frac{\pi}{2})$ for which vectors $\vec{\text{a}}=(\sin\theta)\hat{\text{i}}+(\cos\theta)\hat{\text{j}}$ and $\vec{\text{b}}=\hat{\text{i}}-\sqrt{3}\hat{\text{j}}+2\hat{\text{k}}$ are perpendicular.
→Evaluate the following integrals:$\int\frac{\sec^2\text{x}}{\sqrt{4+\tan^2\text{x}}}\text{ dx}$
→