dy a. Use implicit differentiation to find the derivative dx b. Find the slope of the curve at the given point. sin (y) = 4x# -4; (1,1) dy a. - 16 dx17 3 A spherical balloon is inflating with helium at a rate of 72x min How fast is the balloon's radius increasing at the instant the radius is 3 ft? Write an equation relating the volume of a sphere, V, and the radius of the sphere, r. (Type an exact answer, using It as needed.)a. Determine whether the Mean Value Theorem applies to the function f(x) = -7 + x- on the interval [- 1,2]. b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. a. Choose the correct answer below. O A. No, because the function is differentiable on the interval ( - 1,2), but is not continuous on the interval [- 1,2]. O B. No, because the function is not continuous on the interval [ -1,2], and is not differentiable on the interval ( - 1,2). O C. Yes, because the function is continuous on the interval [ - 1,2] and differentiable on the interval ( - 1,2). O D. No, because the function is continuous on the interval [ - 1,2], but is not differentiable on the interval ( - 1,2).Find the intervals on which f is increasing and the intervals on which it is decreasing. f(x) = - 1+x- Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. O A. The function is increasing on the open interval(s) and decreasing on the open interval(s) (Simplify your answers. Type your answers in interval notation. Use a comma to separate answers as needed.) O B. The function is increasing on the open interval(s) . The function is never decreasing. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) O C. The function is decreasing on the open interval(s) . The function is never increasing. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) O D. The function is never increasing nor decreasing.Perform a first derivative test on the function f(x) = x + 3x -5; [-5,5]. a. Locate the critical points of the given function. b. Use the first derivative test to locate the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). a. Locate the critical points of the given function. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The critical point(s) is/are at x = (Simplify your answer. Use a comma to separate answers as needed.) O B. The function does not have a critical point.Of all rectangles with a perimeter of 30, which one has the maximum area? (Give the dimensions.) The rectangle that has the maximum area has length and width (Simplify your answers.)Use the substitution u = 2x + 3x to evaluate the indefinite integral below. (4x + 3) 12x2 + 3x dx Write the integrand in terms of u. (4x + 3)12x + 3x dx= (2x2+3x) (4x + 3) du