The second derivative of the function f is given by f′′(x)=sin((x^2)/8)−2cosx. The function f has many critical points, two of which are at x=0 and x=6.949. Which of the following statements is true?
a) f has a local minimum at x=0 and at x=6.949.
b) f has a local minimum at x=0 and a local maximum at x=6.949.
c) f has a local maximum at x=0 and a local minimum at x=6.949.
d) f has a local maximum at x=0 and at x=6.949.
f″(x) = sin(x28)−2cos(x)
Let's plug in each of the given critical x-values into the second derivative to see whether it is positive or negative.
f″(0) = sin(028)−2cos(0) = sin(0)−2cos(0) = 0−2(1) = −2 < 0
Since f''(0) < 0 the graph of f is concave down at x = 0 and so a local max occurs at x = 0
f″(6.949) = sin(6.94928)−2cos(6.949) ≈ −1.817 < 0 (assuming x is in radians)
Since f''(6.949) < 0 the graph of f is concave down at x = 6.949 and so a local max occurs at x = 6.949
Here's some more info about this:
f″(x) = sin(x28)−2cos(x)
Let's plug in each of the given critical x-values into the second derivative to see whether it is positive or negative.
f″(0) = sin(028)−2cos(0) = sin(0)−2cos(0) = 0−2(1) = −2 < 0
Since f''(0) < 0 the graph of f is concave down at x = 0 and so a local max occurs at x = 0
f″(6.949) = sin(6.94928)−2cos(6.949) ≈ −1.817 < 0 (assuming x is in radians)
Since f''(6.949) < 0 the graph of f is concave down at x = 6.949 and so a local max occurs at x = 6.949
Here's some more info about this: