Test for Monotonicity about a Point

Q. $f\left(x\right)$ is cubic polynomial with $f\left(2\right)=18$ and $f\left(1\right)=-1.$ Also $f\left(x\right)$ has local maxima at $x=-1$ and $f\text{'}\left(x\right)$ has local minima at $x=0$, then

1. the distance between $(-1,2)$ and $(a,f(a\left)\right)$, where $x=a$ is the point of local minima is $2\sqrt{5}$
2. $f\left(x\right)$ is increasing for $x\in [1,2\sqrt{5}]$
3. $f\left(x\right)$ has local minima at $x=1$
4. the value of $f\left(0\right)=15$

Q. If $f:R\to R$ and $g:R\to R$ are two functions such that $f\left(x\right)+f\text{'}\text{'}\left(x\right)=-xg\left(x\right)f\text{'}\left(x\right)$ and $g\left(x\right)>0\forall x\in R$, then the function $f^2\left(x\right)+\left(f\text{'}\right(x){)}^2$ has

1. a maxima at $x=0$
2. a minima at $x=0$
3. a point of inflexion at $x=0$
4. None of these

Q.

Find the point(s) where the function f(x) = $x^3$ - 3x + 2 is increasing

1. x = 0

2. x = 1

3. x = 2

4. None of these

Q.

A function f is defined by f(x) = 2x – 5. Write down the values of

(i) f(0), (ii) f(7), (iii) f(–3)