Find the relative maxima and relative minima, if any, of the function.

Question

F(x) = 3x⁴ - 4x³ + 7

Relative Maximum (X,Y) = ( )

Relative Minimum (X,Y) = ( )

Answer 1

F(x) = 3x^4 - 4x^3 + 7

F'(x) = 12x^3 -12x^2 = 0

= 12x^2(x -1) = 0

x = 0 and 1

y=7 and 3(1)^4 - 4(1)^3 + 7 = 3-4+7 = 6

(0,7) and (1,6) are the local extrema, or one of them is, 7>6 so

(0,7) would be the likely relative maximum, bu it's is an inflection point, Not a relative maximum. There is no relative maximum

(1,6) is a relative minimum.

F"(x) = 36x^2 -24

F"(0) = 0, so (0,6) is an inflection point not a relative maximum. the 2nd derivative has to be <0 for the point to be a relative maximum.

F"(1) = 36-24 = 12 >0 so (1,6) is a relative minimum