4. (a) Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$. (b) Given that $f'(x) = x^{3} + 12x^{2} + 50x - 11$, find $f''(x)$. (c) Hence, or otherwise, explain why the curve with equation $y = f'(x)$ is strictly increasing for all values of $x$.

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4. (a) Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$ - Scottish Highers Maths - Question 4 - 2017

Question 4

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4. (a) Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$. (b) Given that $f'(x) = x^{3} + 12x^{2} + 50x - 11$, find $f''(x)$. (c) Hence, or otherwise, exp... show full transcript

Worked Solution & Example Answer:4. (a) Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$ - Scottish Highers Maths - Question 4 - 2017

Step 1

Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$

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Answer

To express the quadratic equation in the desired form, we first factor out 3 from the quadratic terms:

$3(x^{2} + 8x) + 50$

Next, we complete the square for the expression inside the parentheses:

Take half of the coefficient of $x$ , which is 8, giving 4. Then square it to get $4^{2} = 16$ .
Add and subtract this square within the parentheses:

$3(x^{2} + 8x + 16 - 16) + 50$ 3. Now rewrite it:

$3((x + 4)^{2} - 16) + 50$ 4. Distribute the 3:

$3(x + 4)^{2} - 48 + 50$ 5. Combine the constants:

$3(x + 4)^{2} + 2$

Thus, the expression in the required form is:

$(ax + b)^{2} + c = 3(x + 4)^{2} + 2$

Step 2

Given that $f'(x) = x^{3} + 12x^{2} + 50x - 11$, find $f''(x)$

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Answer

To find the second derivative $f''(x)$ , we differentiate $f'(x)$ :

Differentiate each term:
- The derivative of $x^{3}$ is $3x^{2}$ .
- The derivative of $12x^{2}$ is $24x$ .
- The derivative of $50x$ is $50$ .
- The derivative of the constant $-11$ is $0$ .

So, combining these, we get:

$f''(x) = 3x^{2} + 24x + 50$

Step 3

Hence, or otherwise, explain why the curve with equation $y = f'(x)$ is strictly increasing for all values of $x$

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Answer

To determine if the curve is strictly increasing, we want to analyze the first derivative $f'(x)$ . A function is strictly increasing where its derivative is positive:

We already found that:

$f''(x) = 3x^{2} + 24x + 50$

This is a quadratic function that opens upwards (since the coefficient of $x^{2}$ is positive). To check if it is always positive, we can evaluate its discriminant:

$D = b^{2} - 4ac = 24^{2} - 4(3)(50)$ $D = 576 - 600 = -24$

Since the discriminant is negative, $f''(x)$ has no real roots and is always positive. Hence, $f'(x)$ is strictly increasing for all values of $x$ .

4. (a) Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$ - Scottish Highers Maths - Question 4 - 2017

Worked Solution & Example Answer:4. (a) Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$ - Scottish Highers Maths - Question 4 - 2017

Express $3x^{2} + 24x + 50$ in the form $(ax + b)^{2} + c$

Given that $f'(x) = x^{3} + 12x^{2} + 50x - 11$, find $f''(x)$

Hence, or otherwise, explain why the curve with equation $y = f'(x)$ is strictly increasing for all values of $x$

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