1. (8pt) Consider the function f(x) = (x + 3)(x + 5)^2 √ (7 − x)...

1. (8pt) Consider the function f(x) = (x + 3)(x + 5)^2 √ (7 − x) whose first and second derivatives are f'(x) = ((x + 5)(139 + 12x − 7x^2))/2 √ (7 − x) , f''(x) = (35x^3 − 225x^2 − 843x + 3481)/ 4(7 − x)^3/2 .

Note: 35x 3 − 225x 2 − 843x + 3481 has three roots: x1 ≈ 7.8835, x2 ≈ −4.3531 and x3 ≈ 2.8982.

(a) (1pt) What is the domain of f(x)? (b) (1pt) What are the x-intercepts and y-intercepts of f(x)? (c) (1pt) What are the critical numbers of f(x)? (d) (1pt) On what open interval(s) is f(x) increasing, and on what open interval(s) f(x) is decreasing? (e) (1pt) What are the x-values of potential points of inflection of f(x)? (f) (1pt) On what open interval(s) is f(x) concave-upward, and on what open interval(s) f(x) is concave-downward? (g) (2pt) Sketch y = f(x). Clearly label the intercepts, extrema, and points of inflection.

Expert Solution

milcah answered 2 years ago

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