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Find the intervals in which f(x)=x^2+2x-...

Find the intervals in which `f(x)=x^2+2x-5` is increasing or decreasing.

Text Solution

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From the graph , ` f' lt 0" for " x in (1, 3) cup (5, 6)` where f decreases.
From the graph,` f' gt 0 " for " x in (0, 1) in (3, 5)` where f increases.
At x = 1 and x = 5, f' changes sign from '+' to '-', so it is the point of local maxima.
At x = 3, f'changes sign from '-' to '+' , so it is the point of local minima.
Slope of tangent to curve f' is '-' for ` x in (0, 2), i.e. f''lt 0`.
Slope of tangent to curve f ' is '+' for ` x in (2, 4), i.e. f'' gt 0`.
That is , f'' changes sign at x = 2 with f''(2) = 0, so x = 2 is the point of inflection.
Further slope of the tangent to the curve f' is '-' for ` x in (4, 6) , i.e. f'' lt 0`.
That is , f'' changes sign at x = 4 again f ''(4) = 0, so x = 4 is the point of inflection.
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Knowledge Check

  • The interval in which y = x^(2) e^(–x) is increasing is

    A
    (`-oo,oo)`
    B
    (-2,0)
    C
    (2,00)
    D
    (0,2)
  • If f(x)=int_(x^(2))^(x^(2)+1)e^(-t^(2))dt , then the interval in which f(x) is increasing, is-

    A
    `(-oo , 0)`
    B
    `(0, oo)`
    C
    `[-2, 2]`
    D
    `[3, 5]`
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