i) What do you mean by the minimum point of a function? ii) The function y = 2x3 + ax2 + bx has critical…

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i) What do you mean by the minimum point of a function? ii) The function y = 2x3 + ax2 + bx has critical points at x=2 and x=1. Find: (i) The values of a and b. (ii) The values of x for which y is min

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a asks:

i) What do you mean by the minimum point of a function?

ii) The function $y = 2x^3 + ax^2 + bx$ where $a$ and $b$ are constant, has critical points at $x=2$ and $x=1$ . Find:
(i) The values of $a$ and $b$ .
(ii) The values of $x$ for which $y$ is minimum.
(iii) The minimum value of $y$ .

Define the minimum point of a function.
A minimum point of a function is a point where the function's value is less than or equal to the values at all nearby points within its domain. At a local minimum, the first derivative of the function is zero (if differentiable), and the second derivative is positive.
Find the values of $a$ and $b$ .
Given the function $y = 2x^3 + ax^2 + bx$ .
Critical points occur where the first derivative $y'$ is equal to zero.

First, find the first derivative $y'$ :
$y' = \frac{d}{dx}(2x^3 + ax^2 + bx)$ $y' = 6x^2 + 2ax + b$
The critical points are given as $x=1$ and $x=2$ . This means $y'=0$ at these points.

For $x=1$ :
$6(1)^2 + 2a(1) + b = 0$ $6 + 2a + b = 0$ $2a + b = -6 \quad \text{(Equation 1)}$
For $x=2$ :
$6(2)^2 + 2a(2) + b = 0$ $6(4) + 4a + b = 0$ $24 + 4a + b = 0$ $4a + b = -24 \quad \text{(Equation 2)}$
Now, solve the system of linear equations for $a$ and $b$ .
Subtract Equation 1 from Equation 2:
$(4a + b) - (2a + b) = -24 - (-6)$ $2a = -18$ $a = \frac{-18}{2}$ $a = -9$
Substitute the value of $a=-9$ into Equation 1:
$2(-9) + b = -6$ $-18 + b = -6$ $b = -6 + 18$ $b = 12$
The values of $a$ and $b$ are $\boxed{a = -9, b = 12}$ .

I) What do you mean by the minimum point of a function? ii) The function y = 2x3 + ax2 + bx has critical points at x=2 and x=1. Find: (i) The values of a and b. (ii) The values of x for which y is min

Step-by-step solution

Define the minimum point of a function.

Find the values of $a$ and $b$ .

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