Questions: On considère la fonction f: R → R définie par f(x)=x^6(1-x)^2 (a) Donnez la liste de tous les points critiques de f. Séparez les valeurs par un point-virgule s'il y en a plus d'une. Inscrivez le mot "rien" (sans les guillemets et tout en minuscules) s'il n'y en a aucune. x= (b) Donnez la liste de tous les minima locaux de f (en suivant la même consigne). x= (c) Donnez la liste de tous les maxima locaux de f (en suivant la même consigne). x=

(a) Find the critical points of the function \( f(x) = x^6(1-x)^2 \).

Compute the first derivative of \( f \).

The first derivative is given by \( f'(x) = x^6(2x - 2) + 6x^5(1 - x)^2 \).

Set the first derivative equal to zero and solve for \( x \).

The critical points are found to be \( x = 0 \), \( x = \frac{3}{4} \), and \( x = 1 \).

The critical points are \\(\boxed{0; \frac{3}{4}; 1}\\).

(b) Identify all local minima of the function \( f \).

Compute the second derivative of \( f \).

The second derivative is \( f''(x) = 2x^6 + 12x^5(2x - 2) + 30x^4(1 - x)^2 \).

Evaluate the second derivative at each critical point to determine local minima.

The second derivative is positive at \( x = 1 \), indicating a local minimum at this point.

The local minima are \\(\boxed{1}\\).

(c) Identify all local maxima of the function \( f \).

Evaluate the second derivative at the other critical points to determine local maxima.

The second derivative is negative at \( x = \frac{3}{4} \), indicating a local maximum at this point.

The local maxima are \\(\boxed{\frac{3}{4}}\\).

The critical points are \\(\boxed{0; \frac{3}{4}; 1}\\). The local minima are \\(\boxed{1}\\). The local maxima are \\(\boxed{\frac{3}{4}}\\).