Question

1. (12) Find the critical points and the extreme max and min values of the function f(x)= 3x* - 4x' on the interval [-2, 2].

Answer 1

Critical point of a function, where either derivative doesn't exist or vanishes at that point.

given fun, , differentiate it w.r.t x

f(0) = 3.4 - 4r

we get,

since given function's derivative exist everywhere. So, to calculate critical point we will put

i.e.$12x^{2}(x-1)=0$

which implies that either x=0 or x=1

thus critical points for given function are 0 and 1

Extreme values of a continuous function exists either at end point of domain or at critical points.

thus for our given functions we will check value of f(x) at end points and also at critical points

therefore extreme max of f(x) exist at x=-2 and value is 80.

extreme min of f(x) exist at x=1 and value is -1