Question

1) On what intervals is f(x) = x^4 - 4x^3 both decreasing and concave up? Clearly show your thinking using sign charts for f' and f". Attach a Desmos graph that confirms your answer. 2) Given the function g(x) = xe^-x, a) find the intervals where g is concave down, and b) find any points on the graph of g where there is a horizontal tangent line. Attach a Desmos graph that confirms your answers. 3) Given the function f(x) = x^5 + 15x^4 + 25, a) use derivatives to determine the critical numbers of f, and use the first derivative test to decide whether a relative maximum, a relative minimum, or neither occurs at each. b) Find the x and y coordinates of the inflection points of f. Attach a Desmos graph, and mark on the graph the critical points and inflection points of f. 4) Use calculus to find the exact global maximum and minimum values of f(x) = sin^2(x) - sin(x) on the interval [0, 2?]. Attach a Desmos graph that confirms your answer.

          1) On what intervals is f(x) = x^4 - 4x^3 both decreasing and concave up? Clearly show your thinking using sign charts for f' and f". Attach a Desmos graph that confirms your answer. 2) Given the function g(x) = xe^-x, a) find the intervals where g is concave down, and b) find any points on the graph of g where there is a horizontal tangent line. Attach a Desmos graph that confirms your answers. 3) Given the function f(x) = x^5 + 15x^4 + 25, a) use derivatives to determine the critical numbers of f, and use the first derivative test to decide whether a relative maximum, a relative minimum, or neither occurs at each. b) Find the x and y coordinates of the inflection points of f. Attach a Desmos graph, and mark on the graph the critical points and inflection points of f. 4) Use calculus to find the exact global maximum and minimum values of f(x) = sin^2(x) - sin(x) on the interval [0, 2?]. Attach a Desmos graph that confirms your answer.

1) On what intervals is f(x) = x^4 - 4x^3 both decreasing and concave up? Clearly show your thinking using sign charts for f' and f". Attach a Desmos graph that confirms your answer. 2) Given the function g(x) = xe^-x, a) find the intervals where g is concave down, and b) find any points on the graph of g where there is a horizontal tangent line. Attach a Desmos graph that confirms your answers. 3) Given the function f(x) = x^5 + 15x^4 + 25, a) use derivatives to determine the critical numbers of f, and use the first derivative test to decide whether a relative maximum, a relative minimum, or neither occurs at each. b) Find the x and y coordinates of the inflection points of f. Attach a Desmos graph, and mark on the graph the critical points and inflection points of f. 4) Use calculus to find the exact global maximum and minimum values of f(x) = sin^2(x) - sin(x) on the interval [0, 2?]. Attach a Desmos graph that confirms your answer.

Added by Matthew M.

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