Determine the constants a and b in order that the function
F(x)=x^3 + ax^2 + bx + 9 , may have
a) A relative maximum at x=-1 and a relative minimum at x=3
b) A relative minimum at x=4 and a point of inflection at x=1
+23254 a) Relative mins and relative max occur where f' = 0.
f(x) = x³ + ax² + bx + 9 ---> f'(x) = 3x² +2ac + b
f'(-1) = 3(-1)² + 2a(-1) + b = 0 f'(3) = 3(3)² + 2a(3) + b = 0
3 - 2a + b = 0 27 + 6a + b = 0
-2a + b = -3 6a + b = -27
Combining them: ---> - ( -2a + b = -3 )
8a = -24
a = -3, b = -9
b) f'(4) = 3(4)² +2a(4) + b = 0 ---> 48 + 8a + b = 0 ---> 8a + b = -48
At inflexion point, f''(x) = 0
f''(x) = 6x + 2a ---> f''(1) = 6(1) + 2a = 0 ---> 2a = -6
a = -3
b = -24
+23254 a) Relative mins and relative max occur where f' = 0.
f(x) = x³ + ax² + bx + 9 ---> f'(x) = 3x² +2ac + b
f'(-1) = 3(-1)² + 2a(-1) + b = 0 f'(3) = 3(3)² + 2a(3) + b = 0
3 - 2a + b = 0 27 + 6a + b = 0
-2a + b = -3 6a + b = -27
Combining them: ---> - ( -2a + b = -3 )
8a = -24
a = -3, b = -9
b) f'(4) = 3(4)² +2a(4) + b = 0 ---> 48 + 8a + b = 0 ---> 8a + b = -48
At inflexion point, f''(x) = 0
f''(x) = 6x + 2a ---> f''(1) = 6(1) + 2a = 0 ---> 2a = -6
a = -3
b = -24
