Question

Find a function y = f(x) having at least three inflection points and sketch its graph. [Note: To find the formula of f(x) start with a suitably chosen f""(x) and integrate f""(x) twice. The second part of the question is a curve sketching problem. Give all details. And try to make your answer unique]"

A) f(x) = x⁴ - 4³ + 6x² - 4x
B) f(x) = sin(x)
C) f(x) = e^x
D) f(x) = 1/x

Answer 1

To find a function with at least three inflection points, we start with the second derivative of the function and integrate it twice. Option A) f(x) = x⁴ - 4³ + 6x² - 4x is a suitable function that can be used. By integrating f''(x), we find f(x) and sketch its graph by plotting (x, y) points and connecting them smoothly.

Step-by-step explanation:

To find a function with at least three inflection points, we need to integrate f''(x) twice. Let's choose option A) f(x) = x⁴ - 4³ + 6x² - 4x.

First, we find f''(x) by taking the second derivative of f(x).

f''(x) = 12x² - 4

Next, we integrate f''(x) twice to find f(x). We integrate f''(x) once to get f'(x) and then integrate again to get f(x).

Finding f'(x):

f'(x) = ∫(12x² - 4) dx

f'(x) = 4x³ - 4x + C₁

Finding f(x):

f(x) = ∫(4x³ - 4x + C₁) dx

f(x) = x⁴ - 2x² + C₁x + C₂

By choosing suitable values for C₁ and C₂, we can find a function with three inflection points. The graph of the function y = f(x) can be sketched by plotting various (x, y) points and connecting them smoothly.