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If f(x) attains a local minimum at x=...

If `f(x)` attains a local minimum at `x=c` , then write the values of `f^(prime)(c)` and `f"(c)` .

Text Solution

Verified by Experts

Given function,
`f(x)`
Given, `x=c`
`Rightarrow f(x) = f(c)`
derivating the function,
`f^(prime)(c)=0`
To be minima,
`f^(prime)prime(c)` > `0`
...
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Knowledge Check

  • f(c) is a minimum value of f(x) if -

    A
    `f'(c) = 0 , f'' (c) gt 0`
    B
    `f'(c) = 0 , f'' (c) lt 0`
    C
    `f ' (c) ne 0 , f''(c) = 0`
    D
    `f '(c) lt 0 , f ''(c) gt 0`

  • Let f(x)=ax^3+bx^2+cx+d then f(x)=3ax^2+2bx+c Also, if (k) = 0 will give maxima/minima. Also, if there are various peak values of a graph then the highest peak value is called the absolute or global maximum and rest are called the local maxima. Similarly absolute or global minimum and local minima too can be defined f(x) is aibic polynomial which has local maximum at x=-1 . If f(-1)=10,f(1)=-6 and f(x) has local minimum at x = 1 Find the distance between the local maximum and local minimum of the curve given by f(x)

    A
    `4sqrt65`
    B
    `7sqrt65`
    C
    `12sqrt35`
    D
    `65sqrt6`

  • Let f(x)=x^(3)+ax^(2)+bx+c , where a, b, c are real numbers. If f(x) has a local minimum at x = 1 and a local maximum at x=-1/3 and f(2)=0 , then int_(-1)^(1) f(x) dx equals-

    A
    `14/3`
    B
    `(-14)/3`
    C
    `7/3`
    D
    `(-7)/3`

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    Let f and g be two differentiable functions defined from R->R^+. If f(x) has a local maximum at x = c and g(x) has a local minimum at x = c, then h(x) = f(x)/g(x) (A) has a local maximum at x = c (B) has a local minimum at x = c (C) is monotonic at x = c 1.a g(x) (D) has a point of inflection at x = c

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