WITH THE GIVEN INPUTS AND OUTPUTS ARE THERE ANY GUARANTEED EXTREMA

Let 𝒇(𝒙) be a polynomial function with the given values. Are there any guaranteed extrema? If so, state where they occur.

Problem 1 :

f(-1) = 0, f(0) = 6 and f(6) = 0

Solution :

By observing the inputs and outputs,

f(0) > f(-1)

f(6) < f(0)

• In the interval [-1, 0] the function must be increasing because considering the outputs 6 is greater than the previous output 0.
• In the interval [0, 6] the function must be decreasing because considering the outputs 0 is lesser than the previous output 6.

The curve changes its path from increasing to decreasing, then there must be guaranteed extrema in the interval [-1, 6].

Problem 2 :

f(0) = 6, f(3) = 2, f(6) = 0 and f(10) = 0

Solution :

By observing the inputs and outputs,

f(3) < f(0)

f(6) < f(3)

f(6) and (10) = 0

• In the interval [0, 3] the function must be decreasing because considering the outputs 2 is lesser than the previous output 6.
• In the interval [3, 6] the function must be decreasing because considering the outputs 0 is lesser than the previous output 2.

The curve doesn't changes its direction in the interval [0, 3], by ploting the points in the grid, we get

As we can draw many possible c urves. We observe there must be maximum or minimum. so, extrema is guaranteed.

Problem 3 :

f(-5) = 0, f(0) = 5 and f(5) = 7

Solution :

By observing the inputs and outputs,

f(0) > f(- 5)

f(5) >