Question

Let \(f(x)\) be the cubic polynomial having one of its roots as \(x=-1\). If \(f(0)=4,f'(0)=0,f^{''}(1)=0\), then...

• A. \(f(x)\) is increasing in \((-\infty ,0)\cup (2,\infty )\)
• B. \(f(x)\) is decreasing in \((-\infty ,0)\cup (2,\infty )\)
• C. \(f(x)\) is increasing in \((0,2)\)
• D. \(f(x)\) is decreasing in \((2,\infty )\)

Hint:

Use the conditions to find the cubic, then study the sign of its derivative.

Answer 1

The Correct Option is A

Step 1: Find the cubic
From $f(0)=4$ and $f'(0)=0$ we get $f(x)=ax^{3}+bx^{2}+4$.
$f''(1)=6a+2b=0$ gives $b=-3a$.
Using $f(-1)=0$: $-a-3a+4=0$, so $a=1,\ b=-3$.

Step 2: Find critical points
$f'(x)=3x^{2}-6x=0\Rightarrow x=0,2$.

Step 3: Test intervals
On $(0,2)$, for example at $x=1$, $f'(1)=-3<0$, so $f$ decreases.
Outside this interval $f'>0$, so $f$ increases in $(-\infty,0)\cup(2,\infty)$.

Final Answer:
Hence f increases in (-infinity, 0) and (2, infinity). \[ \boxed{\text{(A) }f(x)\text{ is increasing in }(-\infty ,0)\cup (2,\infty )} \]