Find the zeros of the function algebraically.

f(x) = 36x^4 − 25x^2

x = (smallest value)
x =
x = (largest value)

(6x^2-5x)(6x^2+5x)=0

x(6x-5)=0 x(6x+5)=0
x=0 (6x-5)=0 x=0 6x+5=0
6x=5 x=-5/6
x=5/6

To find the zeros of the function algebraically, we need to solve the equation f(x) = 0. In this case, the function is given by f(x) = 36x^4 − 25x^2.

Step 1: Factor out the common term.
f(x) = x^2(36x^2 - 25)

Step 2: Set each factor equal to zero and solve for x.

Setting x^2 = 0, we get x = 0.

Next, setting 36x^2 - 25 = 0, we can solve for x by using the difference of squares formula: a^2 - b^2 = (a + b)(a - b).

36x^2 - 25 = (6x)^2 - 5^2
= (6x + 5)(6x - 5)

Setting each factor equal to zero, we get:
6x + 5 = 0, which gives x = -5/6
6x - 5 = 0, which gives x = 5/6

Therefore, the zeros of the function f(x) = 36x^4 − 25x^2 are:
x = 0, x = -5/6, and x = 5/6.

Regarding the smallest and largest values, it seems you are looking for the range of values for x. In this case, the values of x can be any real number, so there is no specific smallest or largest value.