Application of Derivatives

Find the intervals in which the function f given by f(x) = 2x 2 – 3x is (a) strictly increasing (b) strictly decreasing - Wired Faculty

Question

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Find the intervals in which the function f given by f(x) = 2x² – 3x is
(a) strictly increasing (b) strictly decreasing

Solution

f(x) = 2x² – 3x $rightwards double arrow space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 4 straight x minus 3$
Now $straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space space space space space space space space space space rightwards double arrow space space space space 4 straight x minus 3 space equals space 0 space space space space space rightwards double arrow straight x space equals space 3 over 4$
The point $straight x equals 3 over 4$ divides the real line into two disjoint intervals $open parentheses negative infinity comma space 3 over 4 close parentheses space and space open parentheses 3 over 4 comma space infinity close parentheses$ .
$In space space open parentheses negative infinity comma space space space 3 over 4 close parentheses comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 4 straight x minus 3 space space less than space space 0 therefore space space space space space space space space space straight f space is space strictly space decreasing space in space open parentheses negative infinity comma space space 3 over 4 close parentheses. In space open parentheses 3 over 4 comma space infinity space close parentheses comma space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 4 straight x minus 3 space greater than space 0 therefore space space space space straight f space is space strictly space increasing space in space open parentheses 3 over 4 comma space infinity close parentheses.$

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