If f(x) = 2x2 - 3x + 1, find the value of 2f(0) + f(−1) − 3f(f(0)). 2. If 10x + 10y = 2, then the smallest…

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If f(x) = 2x2 - 3x + 1, find the value of 2f(0) + f(−1) − 3f(f(0)). 2. If 10x + 10y = 2, then the smallest possible value of (x+y) is

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Here are the solutions to the problems:

1. If $f(x) = 2x^2 - 3x + 1$ , find the value of $2f(0) + f(-1) - 3f(f(0))$ .

Calculate $f(0)$ .
$f(0) = 2(0)^2 - 3(0) + 1 = 0 - 0 + 1 = 1$
Calculate $f(-1)$ .
$f(-1) = 2(-1)^2 - 3(-1) + 1 = 2(1) + 3 + 1 = 2 + 3 + 1 = 6$
Calculate $f(f(0))$ .
Since $f(0)=1$ , we need to find $f(1)$ .
$f(1) = 2(1)^2 - 3(1) + 1 = 2 - 3 + 1 = 0$
Substitute the calculated values into the expression.
$2f(0) + f(-1) - 3f(f(0)) = 2(1) + 6 - 3(0) = 2 + 6 - 0 = 8$
The value is 8.

The correct option is (B).
$\boxed{\text{B}}$
2. If $\log_{10}x + \log_{10}y = 2$ , then the smallest possible value of $(x+y)$ is
Use the logarithm property $\log_a M + \log_a N = \log_a (MN)$ .
$\log_{10}(xy) = 2$
Convert the logarithmic equation to an exponential equation.
$xy = 10^2$ $xy = 100$
To find the smallest possible value of $x+y$ for positive $x$ and $y$ (since they are arguments of a logarithm), we use the AM-GM inequality, which states that for non-negative numbers $a$ and $b$ , $\frac{a+b}{2} \ge \sqrt{ab}$ .
$\frac{x+y}{2} \ge \sqrt{xy}$
Substitute $xy=100$ into the inequality.
$\frac{x+y}{2} \ge \sqrt{100}$ $\frac{x+y}{2} \ge 10$ $x+y \ge 20$
The equality holds when $x=y$ . If $x=y$ , then $x \cdot x = 100 \implies x^2 = 100$ . Since $x>0$ , $x=10$ . Thus, $y=10$ .
The smallest value of $x+y$ is $10+10=20$ .

The correct option is (C).
$\boxed{\text{C}}$
3. If $y = \sqrt{x} e^x$ , find $\frac{dy}{dx}$ .
Identify the function as a product of two functions
$u(x) = \sqrt{x}$ and $v(x) = e^x$ . We will use the product rule for differentiation: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ .
Find the derivative of $u(x) = \sqrt{x} = x^{1/2}$ .
$\frac{du}{dx} = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$
Find the derivative of $v(x) = e^x$ .
$\frac{dv}{dx} = e^x$
Apply the product rule.
$\frac{dy}{dx} = (\sqrt{x})(e^x) + (e^x)\left(\frac{1}{2\sqrt{x}}\right)$
Factor out $e^x$ and simplify.
$\frac{dy}{dx} = e^x \left(\sqrt{x} + \frac{1}{2\sqrt{x}}\right)$
This matches option (B).

The correct option is (B).
$\boxed{\text{B}}$

Point	x	y
y-intercept	0	1
x-intercept	0.5	0
Minimum	0.75	−0.12
x-intercept	1	0

If f(x) = 2x2 − 3x + 1, find the value of 2f(0) + f(-1) - 3f(f(0)). 2. If 10x + 10y = 2, then the smallest possible value of (x+y) is

Answer

Step-by-step solution

Calculate $f(0)$ .

Calculate $f(-1)$ .

Calculate $f(f(0))$ .

Substitute the calculated values into the expression.

Use the logarithm property $\log_a M + \log_a N = \log_a (MN)$ .

Convert the logarithmic equation to an exponential equation.

To find the smallest possible value of $x+y$ for positive $x$ and $y$ (since they are arguments of a logarithm), we use the AM-GM inequality, which states that for non-negative numbers $a$ and $b$ , $\frac{a+b}{2} \ge \sqrt{ab}$ .

Substitute $xy=100$ into the inequality.

Identify the function as a product of two functions

Find the derivative of $u(x) = \sqrt{x} = x^{1/2}$ .

Find the derivative of $v(x) = e^x$ .

Apply the product rule.

Factor out $e^x$ and simplify.

Try one yourself

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More Mathematics Questions

If f(x) = 2x2 − 3x + 1, find the value of 2f(0) + f(-1) - 3f(f(0)). 2. If 10x + 10y = 2, then the smallest possible value of (x+y) is

Answer

Step-by-step solution

Calculate f(0)f(0)f(0).

Calculate f(−1)f(-1)f(−1).

Calculate f(f(0))f(f(0))f(f(0)).