Identify and classify the stationary points of f using the second derivative test, where f is the function given below
f(x) = x4- 4x3+ 4x2
Given the function above, we need to find its stationary points. Its stationary points occur either at the minima and maxima of the function or at the endpoints. Noticing that the domain of the function is from positive infinity to negative infinity, we proceed to find the maxima and minima of the graph by finding the points at which the tangent like to the graph has slope 0. We will do this by taking the first derivative and equating to zero:
f'(x) = 4x3 - 12x2 + 8x = 0
Simplifying: 0 = 4x(x2 - 3x + 2) = 4x (x - 1)(x - 2)
Hence, the function's stationary points occur at: x = 0, x = 1 and x = 2 Since all points belong to the domain, we proceed to classify them using the second derivative test:Finding the second derivative of the function:f''(x) = 12x2 - 24x + 8 = 4 (3x2 - 6x + 2) Then, to classify the stationary points we need to find the second derivative of the function at each of the points: For x = 0: f''(0) = 8 Since 8 > 0, we conclude that x = 0 is a local minimumFor x = 1: f''(1) = - 4 Since - 4 < 0, we conclude that x = 1 is a local maximum For x = 2: f''(2) = 8 Since 8 > 0, we conclude that x = 2 is a local minimum
Related Maths IB answers
All answers ▸When integrating by parts, how do I decide which part of the integrand is u or f(x) and which dv or g'(x)?
How to find a modulus and argument of w that is a quotient of z1 and z2 such that z1 = 1 + root(3)i and z2 = 1+ i using modulus-argument form?
Let f (x) = sin(x-1) , 0 ≤ x ≤ 2 π + 1 , Find the volume of the solid formed when the region bounded by y =ƒ( x) , and the lines x = 0 , y = 0 and y = 1 is rotated by 2π about the y-axis.
Given that w=x * e^-w use implicit differentiation to show that dw/dx=1/(e^w + x)