7. (a) Express $-6x^2 + 24x - 25$ in the form $p(x + q)^2 + r$ - Scottish Highers Maths - Question 7 - 2019
Question 7
7. (a) Express $-6x^2 + 24x - 25$ in the form $p(x + q)^2 + r$.
(b) Given that $f(x) = -2x^3 + 12x^2 - 25x + 9$, show that $f'(x)$ is strictly decreasing for all $x... show full transcript
Worked Solution & Example Answer:7. (a) Express $-6x^2 + 24x - 25$ in the form $p(x + q)^2 + r$ - Scottish Highers Maths - Question 7 - 2019
Step 1
Show that $f'(x)$ is strictly decreasing for all $x \\in \mathbb{R}$
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Answer
Given the function:
f(x)=−2x3+12x2−25x+9
Differentiate: Calculate the first derivative:
f′(x)=dxdf=−6x2+24x−25
Analyze the quadratic: To determine whether f′(x) is strictly decreasing, we examine the derivative f′(x), which is a quadratic function. The leading coefficient is −6<0, indicating that it opens downwards.
Find the vertex: The vertex can be found using:
x=−2ab=−2(−6)24=2
We can calculate:
f′(2)=−6(2)2+24(2)−25=−24+48−25=−1<0. So f′(x) is less than zero at the vertex.
Sign of f′(x): Since the parabola opens downwards and the vertex is at a maximum of −1<0, we conclude that:
f′(x)<0 for all xinR, thus proving that f′(x) is strictly decreasing.
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