f(x)=12x^5+45x^4–80x^3+2
the question asks to find the inflection points but i don't seem to understand how to find this even by taking the second derivative.
To find the inflection points of a function, we need to analyze the concavity of the function. In other words, we will examine the behavior of the second derivative of the function.
To find the second derivative of the given function f(x) = 12x^5 + 45x^4 - 80x^3 + 2, we need to take the derivative twice.
First, let's find the first derivative:
f'(x) = d/dx (12x^5 + 45x^4 - 80x^3 + 2)
To take the derivative, we use the power rule, which states that the derivative of x^n with respect to x is n*x^(n-1).
Applying the power rule, we have:
f'(x) = 60x^4 + 180x^3 - 240x^2
Now, let's find the second derivative:
f''(x) = d/dx (60x^4 + 180x^3 - 240x^2)
Applying the power rule again, we have:
f''(x) = 240x^3 + 540x^2 - 480x
To find the inflection points, we need to find the values of x for which the second derivative equals zero or is undefined. In other words, we need to solve the equation f''(x) = 0.
Setting f''(x) = 0, we have:
240x^3 + 540x^2 - 480x = 0
We can factor out an x from this equation to simplify it:
x(240x^2 + 540x - 480) = 0
Now, we need to solve the quadratic equation inside the parentheses:
240x^2 + 540x - 480 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 240x^2 + 540x - 480 = 0, the coefficients are:
a = 240, b = 540, c = -480
Plugging these values into the quadratic formula, we get:
x = (-540 ± √(540^2 - 4*240*(-480))) / (2*240)
Simplifying this equation, we get two values of x:
x = (-540 ± √(291600 + 460800)) / 480
x = (-540 ± √(752400)) / 480
x = (-540 ± 870) / 480
From these solutions, we get two potential inflection points:
x = (-540 + 870) / 480 = 330 / 480 = 11/16
x = (-540 - 870) / 480 = -1410 / 480 = -47/16
To conclude, the potential inflection points of the function f(x) = 12x^5 + 45x^4 - 80x^3 + 2 are x = 11/16 and x = -47/16. However, to determine if these points are indeed inflection points, we need to perform further analysis by finding the behavior of the function around these points using the first or second derivative tests.