How many points of inflexion dose the function y= x^4 have
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y'=4x^3
y"=12x^2
y"'=24x
the degree of the last equation is one, son one inflection point.
To find the number of points of inflection for the function f(x) = x^4, we need to determine where the concavity changes.
Step 1: Find the second derivative of the function f(x):
f(x) = x^4
f'(x) = 4x^3
To find the second derivative, differentiate f'(x) with respect to x:
f''(x) = (d/dx)(4x^3)
= 12x^2
Step 2: Set the second derivative equal to zero and solve for x:
12x^2 = 0
Dividing both sides by 12, we get:
x^2 = 0
Taking the square root of both sides, we find:
x = 0
This means that the second derivative is zero at x = 0.
Step 3: Determine the concavity:
To the left of x = 0, the second derivative is positive (+12x^2), indicating concave up.
To the right of x = 0, the second derivative is negative (-12x^2), indicating concave down.
Since the concavity changes at x = 0, we have one point of inflection.
In conclusion, the function y = x^4 has one point of inflection.