F(x) = 1/2(x+3)(X-7)
Step 1 ; expand the function
F(x)= 1/2(x²-7x+3x-21)
F(x) = 1/2(x² - 4x-21)
F(x) = 1/2x² - 2x-21/2
Step 2 : Take the second derivative of F(x)
This means you are to differentiate F(X) twice
[tex]\begin{gathered} F(x)=\frac{1}{2}x^2-2x-\frac{21}{2} \\ \text{First derivative is} \\ F^!(x)\text{=x-2} \\ F^{!!}(x)=1 \\ \text{the second derivative =1} \end{gathered}[/tex]The second derivative is greater than 0, so it is a minimum point
Put x=1 in F(x) to find the value
[tex]\begin{gathered} f(x)=\frac{1}{2}(1)^2_{}-\text{ 2(1)-}\frac{21}{2} \\ f(x)=\frac{1}{2}-2-\frac{21}{2} \\ f(x)=-2-\frac{20}{2} \\ f(x)\text{ =-12} \end{gathered}[/tex]The minimum of the quadratic equation is -12
