Question

Find the local and absolute extreme values of the function on the given interval. Also

specify the intervals where function is increasing or decreasing

(i) 𝑓(𝑥) = (𝑥²+ 𝑥 + 1)²

Answer 1

$f'(x)=\frac{\mathrm{d}}{\mathrm{d} x}\left[\left(x^{2}+x+1\right)^{2}\right]\\ f'(x)=2\left(x^{2}+x+1\right) \cdot \frac{\mathrm{d}}{\mathrm{d} x}\left[x^{2}+x+1\right]\\ f'(x)=2\left(x^{2}+x+1\right)\left(\frac{\mathrm{d}}{\mathrm{d} x}\left[x^{2}\right]+\frac{\mathrm{d}}{\mathrm{d} x}[x]+\frac{\mathrm{d}}{\mathrm{d} x}[1]\right)\\ f'(x)=2\left(x^{2}+x+1\right)(2 x+1+0)\\ f'(x)=2(2 x+1)\left(x^{2}+x+1\right)$

At $f'(x)=0,$

$$2(2 x+1)\left(x^{2}+x+1\right)=0$$

which implies,

$$x=-\frac{1}{2}$$

Put the value of $x$ into the given function, we have:

$\text{Local Minimum } = (x,f(x)) = (-\frac{1}{2},\frac{9}{16})$

The function has no absolute extreme value.

The interval where the function is increasing or decreasing is thus:

$\text { Decreasing: }-\infty<x<-\frac{1}{2}\\ \text { Increasing: }-\frac{1}{2}<x<\infty$