Determine the critical points of the function f(x)=x^4-6x^2+8x+10 and state their nature

Answer 1

To find the critical points of a function, we first need to find its derivative, set it equal to zero, and solve for x.

Taking the derivative of f(x) = x^4 - 6x^2 + 8x + 10:
f'(x) = 4x^3 - 12x + 8

Setting f'(x) equal to zero and solving for x:
4x^3 - 12x + 8 = 0

To solve this equation for x, we can factor out a common factor of 4:
4(x^3 - 3x + 2) = 0

Now, we can solve the cubic equation x^3 - 3x + 2 = 0. However, this equation does not have any rational roots, so we need to use other methods, such as synthetic division or a graphing calculator, to find its roots. By using a graphing calculator or any other method, we find that the roots are x ≈ -1.62 and x ≈ -0.37.

Now, we have two critical points: x = -1.62 and x = -0.37. To determine their nature, we can analyze the concavity of the function using the second derivative test.

Taking the derivative of f'(x) = 4x^3 - 12x + 8:
f''(x) = 12x^2 - 12

Plugging the critical points into f''(x):
f''(-1.62) = 12(-1.62)^2 - 12 ≈ -37.166
f''(-0.37) = 12(-0.37)^2 - 12 ≈ -12.372

Since f''(-1.62) is negative, the critical point x = -1.62 corresponds to a local maximum.

Since f''(-0.37) is also negative, the critical point x = -0.37 corresponds to a local maximum.

Therefore, the critical points of f(x) = x^4 - 6x^2 + 8x + 10 are x = -1.62 (local maximum) and x = -0.37 (local maximum).