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If f^(11)(x)lt0(gt0) on an interval (a,b...

If `f^(11)(x)lt0(gt0)` on an interval (a,b) then the curve y=f(x) on this interval is convex (concave) i.e it is below (above) any of its tangent lines If `f^(11)(x_(0))=0` or does not exist and the second derivative changes sign when passing through the point `x_(0)` then the point `(x_(0),f(x))` is the = point of inflection of the curve y=f(x)
If `y=x^(4)+x^(3)-18x^(2)+24x-12` then

A

`(-2,-24)` is a point of inflection

B

`(-2,3//2)` is a point of inflection

C

`((3)/(2),-8(1)/(16))` is a point of inflection

D

y is convex on `((3)/(2),oo)`

Text Solution

Verified by Experts

The correct Answer is:
C
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Knowledge Check

  • If f^(11)(x)lt0(gt0) on an interval (a,b) then the curve y=f(x) on this interval is convex (concave) i.e it is below (above) any of its tangent lines If f^(11)(x_(0))=0 or does not exist and the second derivative changes sign when passing through the point x_(0) then the point (x_(0),f(x)) is the = point of inflection of the curve y=f(x) If y=xsin(logx) then

    A
    y has only teo points of inflection
    B
    y has only 4 points of inflection
    C
    `(pi)/(4)` is the only point of inflection
    D
    y has infinite number of points of inflection

  • If f^(11)(x)lt0(gt0) on an interval (a,b) then the curve y=f(x) on this interval is convex (concave) i.e it is below (above) any of its tangent lines If f^(11)(x_(0))=0 or does not exist and the second derivative changes sign when passing through the point x_(0) then the point (x_(0),f(x)) is the = point of inflection of the curve y=f(x) y=x^(4)+ax^(3)+(3)/(2)x^(2)+1 is concave along the entire number scale then

    A
    `|a||ge1`
    B
    `|a|le1`
    C
    `|a|le2`
    D
    `|a|gt2`

  • In the interval (0,pi), f(x)=sinx-x is

    A
    decreasing
    B
    increasing
    C
    stationary
    D
    ocilating

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