how do you find the inflection points of a polynomial such as f(x)=12x^5+45x^4–80x^3+2 .
Look for points where the second derivative is zero. The first derivative (slope) must also change sign there.
You will have to solve this cubic equation to get the inflection point.
240 x^3 + 540 x^2 - 480 x = 0
You can factor this down to
x (12x^2 + 27 x -20) = 0
x = 0 is one of the inflection points; there appear to be two others but the quadratic term does not factor easily.
To find the inflection points of a polynomial function, such as f(x) = 12x^5 + 45x^4 - 80x^3 + 2, you need to follow these steps:
Step 1: Determine the second derivative of the function. The second derivative represents the rate at which the slope of the function is changing.
f''(x) = d^2/dx^2 (12x^5 + 45x^4 - 80x^3 + 2)
Step 2: Simplify the second derivative by differentiating each term of the function and combining like terms.
f''(x) = (240x^3) + (360x^2) - (480x^2)
Simplifying further:
f''(x) = 240x^3 + 360x^2 - 480x^2
f''(x) = 240x^3 - 120x^2
Step 3: Set the second derivative equal to zero because inflection points occur where the slope changes sign. Solve for x.
240x^3 - 120x^2 = 0
Factoring out the common factor:
x^2(240x - 120) = 0
Applying the zero-product property, we get:
x^2 = 0 or 240x - 120 = 0
Solving the first equation, x^2 = 0, we find that x = 0.
Step 4: Solving the second equation, 240x - 120 = 0, you will find the value of x:
240x - 120 = 0
240x = 120
x = 120/240
x = 1/2
Therefore, the possible inflection points of the function f(x) = 12x^5 + 45x^4 - 80x^3 + 2 are x = 0 and x = 1/2.