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Question
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
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Solution
\[\text { A function f(x) is said to be increasing on } \left[ a, b \right] \text { if it is increasing at x = a and x = b } . \]
\[\text { Here, } \]
\[f\left( x \right) = x^2 - 6x + 3\]
\[f'\left( x \right) = 2x - 6\]
\[ \Rightarrow f'\left( x \right) = 2\left( x - 3 \right)\]
\[\text { Now, } f'\left( 4 \right) = 2\left( 4 - 3 \right)\]
\[ = 2\]
\[ \therefore f'\left( 4 \right) > 0 \]
\[\text { So, f(x) is increasing on x} = 4 \]
\[\text { &, }f'\left( 6 \right) = 2\left( 6 - 3 \right)\]
\[ = 6\]
\[ \therefore f'\left( 6 \right) > 0 \]
\[\text { So, f (x) is increasing on x } = 6 \]
\[\text { Hence,}f\left( x \right)\text { is increasing on } [4, 6].\]
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