Find all relative extrema. Use the second derivative test where applicable.
(a) f(x)=√x2+1
(b) f(x)=cos x -x
(a) "f(x)=\\sqrt{x^2+1}=(x^2+1)^{1\\over 2}"
"f'(x)={1\\over 2}(x^2+1)^{-{1\\over 2}}.2x={x\\over \\sqrt {x^2+1}}"
Equate "f'(x)" to to get the value of "x"
"f'(x)={1\\over 2}(x^2+1)^{-{1\\over 2}}.2x={x\\over \\sqrt {x^2+1}}=0 \\implies x=0"
"f'(x)\\lt 0" for "x\\lt 0" and "f'(x)\\gt0" for "x\\gt 0" "\\implies" at "x=0" there is a minimum.
Because "f(x)" was decreasing till it reaches a minimum at "x=0" and then started increasing ,because the derivative is positive after "x=0"
(b) "f(x)=cos( x) -x"
"f'(x)=-sin(x)-1"
Equate "f'(x)" to to get the value of "x"
"f'(x)=-sin(x)-1 =0 \\implies x={3\\pi \\over 2}"
"f''(x)=-cos(x)"
"f''({3\\pi \\over 2})=-cos({3\\pi \\over 2})=0 \\implies" we have an inflection point at "x={3\\pi \\over 2}"
so,No extrema anywhere
Comments
Leave a comment