Answer:
The local maximum is at x = 0 and local minimum is at x = 1
Step-by-step explanation:
f(x) = 6 + 12x² - 8x³
The second derivative test is as follows:
When a function's slope is zero at x, and the second derivative at x is:
i) less than 0, it is a local maximum
ii) greater than 0, it is a local minimum
iii) equal to 0, then the test fails
[tex]f(x)=6+12x^2-8x^3\\\\\frac{df(x)}{dx} =24x-24x^2\\\\24x-24x^2=0\\24x(1-x)=0\\\\x=0\ and\ x=1[/tex]
[tex]The\ second\ derivative\ is:\\\\f"(x)=24-48x\\\\At\ x=0\\\\f"(0)=24-48(0)=24 >0. (It\ is\ a\ local\ maximum)\\\\At\ x=1\\\\f"(1)=24-48(1)=-24<0. (It\ is\ a\ local\ minimum)[/tex]
The local maximum is at x = 0 and local minimum is at x = 1
