Explanation Given: The function is f ( x ) = 5 − x 2 . Calculation: Calculate f ′ ( x ) as follows. f ′ ( x ) = d d x ( 5 − x 2 ) = 5 − x 2 ( ln 5 ) ( − 2 x ) = − ( 2 x ln 5 ) 5 − x 2 Evaluate f ″ ( x ) from f ′ ( x ) . f ″ ( x ) = d d x ( − ( 2 x ln 5 ) 5 − x 2 ) = − 2 ln 5 ( 5 − x 2 ) + ( − 2 ln 5 ) x ( ln 5 ) ( 5 − x 2 ) ( − 2 x ) = 5 − x 2 ( 2 ln 5 ) ( 2 x 2 ln 5 − 1 ) Thus, the value of f ″ ( x ) is 5 − x 2 ( 2 ln 5 ) ( 2 x 2 ln 5 − 1 ) . For the curve to be concave upward, f ″ ( x ) > 0 5 − x 2 ( 2 ln 5 ) ( 2 x 2 ln 5 − 1 ) > 0 2 x 2 ln 5 − 1 > 0 2 x 2 ln 5 > 1 x 2 > 1 2 ln 5 x > 1 2 ln 5 or x < − 1 2 ln 5 Therefore, the curve is concave upward on the intervals ( − ∞ , − 1 2 ln 5 ) and ( 1 2 ln 5 , ∞ ) . For the curve to be concave downward, f ″ ( x ) < 0 5 − x 2 ( 2 ln 5 ) ( 2 x 2 ln 5 − 1 ) < 0 2 x 2 ln 5 − 1 < 0 2 x 2 ln 5 < 1 x 2 < 1 2 ln 5 − 1 2 ln 5 < x < 1 2 ln 5 Therefore, the curve is concave downward on the interval ( − 1 2 ln 5 , 1 2 ln 5 )

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Chapter 13.3, Problem 50E

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To find: The inflection points and to determine where the given curve is concave upward or downward.

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Chapter 13.3, Problem 49E

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Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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The inflection points and to determine where the given curve is concave upward or downward.

Solution Summary: The author explains that the curve is concave upward on the intervals (-infty,-

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