Question

On what intervals is the graph of f(x)=x^3-2x^2+5x+1 increasing?

Answer 1

$f(x)=x^3-2x^2+5x+1\\ f'(x)=3x^2-4x+5\\ 3x^2-4x+5=0\\\Delta=(-4)^2-4\cdot3\cdot5=16-60=-44$
$\Delta0 \Rightarrow$ the graph of the parabola is above the x-axis, so the derivative is always positive and therefore the initial function is increasing in its whole domain.

$f(x)=0.5x^2-6\\ f'(x)=x$
The function is decreasing when its first derivative is negative. The first derivative of this function is negative for $x$ so for $x\in(-\infty,0)$ the function is decreasing.

$f(x)=\dfrac{x+1}{x-1}\qquad(x\not=1)\\ f'(x)=\dfrac{x-1-(x+1)}{(x-1)^2}=-\dfrac{2}{(x-1)^2}$
The function is increasing when its first derivative is positive. The first derivative of this function is always negative therefore this function is never increasing.