Find the intervals in which the function \(f\) given by \(f(x) = 2x^2 − 3x\) is (a) strictly increasing (b) strictly decreasing
Solution and Explanation
\(f'(x)=4x-3\)
\(f'(x)=0\implies x=\frac{3}{4}\)
Now, the point divides the real line into two disjoint intervals i.e., \((-∞,\frac{3}{4})\)and\((\frac{3}{4},∞)\)In interval \((-∞,-2)\) and \((3,∞),f'(x)\) is positive while in interval \((-2,3), f'(x)\) is negative.
Hence, the given function \((f)\) is strictly increasing in intervals \((-∞,-2)\) and \((3,∞)\), while function \((f)\) is strictly decreasing in interval \((−2, 3)\).
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Notes on Applications of Derivatives
Concepts Used:
Increasing and Decreasing Functions
Increasing Function:
On an interval I, a function f(x) is said to be increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≤ f(y)
Decreasing Function:
On an interval I, a function f(x) is said to be decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≥ f(y)
Strictly Increasing Function:
On an interval I, a function f(x) is said to be strictly increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) < f(y)
Strictly Decreasing Function:
On an interval I, a function f(x) is said to be strictly decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) > f(y)
Graphical Representation of Increasing and Decreasing Functions
