Question

Function $f\left(x\right)=x^4-\frac{x^3}{3}$is

• A. Increasing for $x>\frac{1}{4}$ and decreasing for $<\frac{1}{4}$
• B. Increasing for every value of x
• C. Decreasing for every value of x
• D. None of the above

Answer 1

The correct option is A

Increasing for $x>\frac{1}{4}$ and decreasing for $<\frac{1}{4}$

Explanation for the correct option:

Find the extrema of the given function:

$f\left(x\right)=x^4–\frac{x^3}{3}$

Differentiate it with respect to $x$

$f\text{'}\left(x\right)=4x^3–x^2$

For increasing, $4x^3–x^2>0$

$\Rightarrow$ $x^2(4x–1)>0$

$\Rightarrow$ $4x-1>0$

$\Rightarrow$ $x>\frac{1}{4}$

For decreasing, $4x^3–x^2<0$

$\Rightarrow$ $x^2(4x–1)<0$

$\Rightarrow$ $4x-1<0$

$\Rightarrow$ $x<\frac{1}{4}$

$\therefore$The function is increasing for $\mathit{x}\mathbf{}\mathbf{>}\mathbf{}\frac{\mathbf{1}}{\mathbf{4}}$ and decreasing for $\mathit{x}\mathbf{}\mathbf{<}\mathbf{}\frac{\mathbf{1}}{\mathbf{4}}$

Hence, Option ‘A’ is Correct.