Question

Consider the function $f\left(x\right)=\frac{x^3}{4}-\sin \pi x+3$

• A. $f\left(x\right)$ does not attain value within the interval $[-2,2]$
• B. $f\left(x\right)$ takes on the value $2\frac{1}{3}$ in the interval $[-2,2]$
• C. $f\left(x\right)$ takes on the value $3\frac{1}{4}$ in the interval $[-2,2]$
• D. $f\left(x\right)$ takes no value $p,1<p<5$ in the interval $[-2,2]$

Answer 1

Answer: (B), (C)

The correct options are
B $f\left(x\right)$ takes on the value $2\frac{1}{3}$ in the interval $[-2,2]$
C $f\left(x\right)$ takes on the value $3\frac{1}{4}$ in the interval $[-2,2]$

$f(-2)=1$ and $f\left(2\right)=5$ and $f$ is continuous in $[-2,2].$

So, by using intermediate value theorem we can say that function $f\left(x\right)$ takes all values between $1$ to $5.$
$\Rightarrow 2\frac{1}{3}$ and $3\frac{1}{4}$ lie in $[1,5]$