Consider the function f(x)=x^2-2ax-3x+a^2+3a, a-1 ≤ x ≤ a+3, a>0
The absolute maximum value is __________
and this occurs at equals _________
The absolute minimum value is _____________
and this occurs at equals ____________
Consider the function f(x)=x^2-2ax-3x+a^2+3a, a-1 ≤ x ≤ a+3, a>0
The absolute maximum value is __________
and this occurs at equals _________
The absolute minimum value is _____________
and this occurs at equals ____________
$\displaystyle f(x)=x^2-2ax-3x+a^2+3a$...then$\displaystyle f'(x)=2x-2a-3$...find the zeros...$\displaystyle x=\frac{2a+3}{2}$....it is undefined nowhere...and the original funtion is defined everywhere...off to the second derivative test..$\displaystyle f''(x)=2$...so it is always positive so ...$\displaystyle x=\frac{2a+3}{2}$ is a relative min...now we just need to check the endpoints..$\displaystyle f(a-1)=3a+1$...$\displaystyle f(a+3)=3a+9$...and $\displaystyle f(\frac{2a+3}{2})=a^2+9a+9$...and even though its a relative min the max occurs at $\displaystyle x=\frac{2a+3}{2}$ and the min at $\displaystyle x=a-1$