In the following problems use the function: \( f(x) = x^4 + x^3 \)A. Find the critical numbers of \( f \).B. Determine the intervals where \( f \) is increasing or decreasing.C. Determine any local minima or maxima of \( f \).

The given function is f(x)=x4+x3. a) Find the critical points of f(x)=x4+x3. The derivative of f(x)=x4+x3 is, f'(x)=ddxx4+x3=4x3+3x2 Equate the derivative to zero and obtain the critical points as follows. f'(x)=04x3+3x2=0x24x+3=0x=0, x=-34 Therefore, the critical points of f(x)=x4+x3 are x=0 and x=-34. b) Find the intervals where the function is increasing or decreasing as shown below. The domain of the function f(x)=x4+x3 is -∞,∞. Using the critical points and domain of the function, create three intervals as -∞,-34,-34,0 and 0,∞.Take a test point from each of the above intervals and check whether the function is increasing or decreasing in that interval as shown below. For -∞,-34, consider the point x=-1. f'(-1)=4-13+3-12=-4+3=-1 Since f'(-1)<0, the function is decreasing in the interval -∞,-34. For -34,0, consider the point x=-0.5. f'(-0.5)=4-0.53+3-0.52=-0.5+0.75=0.25 Since f'(-0.5)>0, the function is increasing in the interval -34,0. For 0,∞, consider the point x=1. f'(1)=413+312=4+3=7 Since f'(1)>0, the function is increasing in the interval 0,∞. c) The function is decreasing in the interval -∞,-34 and is increasing in the interval -34,0. That implies, there exists a local minimum at the point x=-34. At x=-34, f(x)=-344+-343=81256-2764=81-108256=-27256 The function is increasing in the interval -34,0 and in the interval 0,∞. That implies, there exists a saddle point x=0. Therefore, there is no local maximum.Hence, Local minimum : -34,-27256 Local maximum : DNE