Question

已知二次函数y=f(x)的最大值等于13,且f(3)=f(1)=5,求f(x)解析式

Answer 1

f(3)=f(1)=5，最大值等于13说明x^2的因数为-
且中轴为x = 2
则f（x） = m(x-2)^2 +13 ,m<0
则f(1) = m+13 = 5 m=-8
f(x) =-8(x-2)^2 +13

Answer 2

let
f(x)= ax^2+ bx + c

f(3)= 9a+3b+c = 5 --(1)
f(1)= a+ b +c = 5 ---(2)

(1)-(2)
8a+2b=0
b= -4a ---(3)

f'(x)= 2ax+b
f'(x)=0 => x=-b/2a
f(-b/2a)= f(2)
= 4a+2b+c = 13 ---_(4)

(1)- (4)
5a+b= -8
5a-4a=-8
a = -8
b = 32
c = -19

f(x)= -8x^2+32x-19 #