Question

Find the intervals in which√(25−4x^2) is strictly increasing or strictly decreasing.

Answer 1

we have to find the interval in which √(25 - 4x²) is strictly increasing or strictly decreasing.

solution : here function, f(x) = √(25 - 4x²)

differentiating f(x) w.r.t x we get,

f'(x) = d[√(25 - 4x²)]/dx

= 1/2√(25 - 4x²) × (-8x)

= -4x/√(25 - 4x²)

at f'(x) = 0

x = 0

now f(x) = 0,

⇒√(25 - 4x²) = 0

⇒x = 5/2 , -5/2

case 1 : 0 < x < 5/2 ⇒ f'(x) < 0

so function is strictly decreasing in the interval (0, 5/2).

case 2 : -5/2 < x < 0 ⇒f'(x) > 0

so function is strictly increasing in the interval (-5/2, 0).

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