Question

Find the minimum and maximum value of the function $f\left(x\right)={(2x-1)}^2+3$.

Answer 1

$f\left(x\right)$ has a maxima or minima at $x$ when $\frac{d\left(f\right(x\left)\right)}{dx}=0$

Now, $f\left(x\right)={(2x-1)}^2+3$

Consider $\frac{d\left(f\right(x\left)\right)}{dx}=0$

$⟹2(2x-1)\left(2\right)=0$

$⟹8x-4=0$

$⟹x=\frac{1}{2}$

$\frac{d^2\left(f\right(x\left)\right)}{dx}=8>0$

$\Rightarrow f\left(x\right)$ has a minima at $x=\frac{1}{2}$

The value of minima $=f\left(\frac{1}{2}\right)={\left(2\right(\frac{1}{2})-1)}^2+3=3$