Answer to Question #286171 in Calculus for Abdullah

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:

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