Question

Consider the function $f\left(x\right)=0.75x^4-x^3-9x^2+7$.

Consider the following statements:

$1$. The function attains local minima at $x=-2$ and $x=3$.
$2$. The function increases in the interval $(-2,0)$.
Which of the above statements is/are correct?

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ nor $2$

Answer 1

The correct option is C Both $1$ and $2$

$f\left(x\right)=0.75x^4-x^3-9x^2+7$

Now, $\frac{df\left(x\right)}{dx}=0$

$⟹3x^3-3x^2-18x=0$

$⟹x=0,-2,3$

$f^{\prime \prime }\left(x\right)=9x^2-6x-18f^{\prime \prime }\left(0\right)=-18<0f^{\prime \prime }(-2)=30>0f^{\prime \prime }\left(3\right)=45>0$

Thus, the function attains local minima at $x=-2$ and $x=3$

Also, from the figure the function increases in the interval $(-2,0)$

Hence, both the statements are correct.