If y=f(x) and the function is known to attain a maximum point at x0, then at the point x0a.f’(x0) <0b.f’(x0) >0c.f’’(x0) >0d.f’’(x0) <0
Question
If y=f(x) and the function is known to attain a maximum point at x0, then at the point x0
a. f’(x0) < 0
b. f’(x0) > 0
c. f’’(x0) > 0
d. f’’(x0) < 0
Solution
The derivative of a function gives us the slope of the tangent line at a given point. At a maximum point, the slope of the tangent line is zero. Therefore, f'(x0) is neither less than 0 (a) nor greater than 0 (b).
The second derivative of a function gives us the concavity of the function. If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down. At a maximum point, the function is concave down. Therefore, f''(x0) < 0 (d).
So, the correct answers are not a, not b, and d.
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