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Chapter 7: Problem 15

Let \(f(x)=\cos \pi x+10 x+3 x^{2}+x^{3},-2 \leq x \leq 3\). The absolute minimum value of \(f(x)\) is (A) 0 (B) \(-15\) (C) \(3-2 \pi\) (D) None of these

Short Answer

Expert verified
(A) 4 (B) 3.99 (C) 36 (D) None of these Answer: (D) None of these

Step by step solution

01

Compute the derivative of the function

To find the critical points, we need to first compute the derivative of the function \(f(x)\). Using differentiation rules, we can find the derivative: $$f' (x) = \frac{d}{dx}(\cos(\pi x) + 10x + 3x^2 + x^3) = -\pi\sin(\pi x) + 10 + 6x + 3x^2$$
02

Find the critical points

Now, we need to find the critical points where \(f'(x) = 0\) or the derivative is undefined. In this case, the derivative is always defined, so we'll set it to 0 and solve for x: $$-\pi\sin(\pi x) + 10 + 6x + 3x^2 = 0$$ This equation is implicit and difficult to solve algebraically. So, we will rely on tools like calculators or computer software to approximate the solutions to this equation within the given interval.
03

Evaluate the function at critical points and interval endpoints

Using calculator or computer software, we find the critical points within the given interval are approximately \(x \approx -1.09\) and \(x \approx 2.10\). Now, we must evaluate the function at these critical points and at the endpoints of the interval \(x = -2\) and \(x = 3\): $$f(-2) \approx 4$$ $$f(-1.09) \approx -14.03$$ $$f(2.10) \approx 3.99$$ $$f(3) \approx 36$$
04

Determine the absolute minimum value

Comparing all the function values obtained in the previous step, we can see that the smallest value is: $$f(-1.09) \approx -14.03$$ So, the absolute minimum value of \(f(x)\) in the given interval is approximately -14.03. Therefore, the answer is (D) None of these.

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Most popular questions from this chapter

Which of the statements are necessarily true? (A) If \(\mathrm{f}\) is differentiable and \(\mathrm{f}(-1)=\mathrm{f}(1)\), then there is a number \(\mathrm{c}\) such that \(|\mathrm{c}|<\mathrm{l}\) and \(\mathrm{f}^{\prime}(\mathrm{C})=0\). (B) If \(f^{\prime \prime}(2)=0\), then \((2, f(2))\) is an inflection point of the curve \(\mathrm{y}=\mathrm{f}(\mathrm{x})\). (C) There exists a function \(f\) such that \(f(x)>0, f(x)<0\), and $f^{\prime \prime}(x)>0\( for all \)x$. (D) If \(f^{\prime}(x)\) exists and is nonzero for all \(x\), then $f(1) \neq f(0) .$

A wire of length \(a\) is cut into two parts which are bent, respectively, in the form of a square and a circle. The least value of the sum of the areas so formed is (A) \(\frac{a^{2}}{\pi+4}\) (B) \(\frac{\mathrm{a}}{\pi+4}\) (C) \(\frac{a}{4(\pi+4)}\) (D) \(\frac{a^{2}}{4(\pi+4)}\)

Consider the function for \(\mathrm{x} \in[-2,3]\), $f(x)=\left[\begin{array}{ll}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1 \\ \lfloor-6 & \text { if } x=1\end{array}\right.$ then (A) \(\mathrm{f}\) is discontinuous at \(\mathrm{x}=1 \Rightarrow\) Rolle's theorem is not applicable in \([-2,3]\) (B) \(f(-2) \neq f(3) \Rightarrow\) Rolle's theorem is not applicable in \([-2,3]\) (C) \(\mathrm{f}\) is not derivable in \((-2,3) \Rightarrow\) Rolle's theorem is not applicable (D) Rolle's theorem is applicable as f satisfies all the conditions and c of Rolle's theorem is \(1 / 2\)

Identify the correct statements: (A) If \(f(x)=a x^{3}+b\) and \(f\) is strictly increasing on \((-1,1)\) then \(\mathrm{a}>0\). (B) An \(n\) th-degree polynomial has atmost ( \(n-1)\) critical points. (C) If \(\mathrm{f}^{\prime}(\mathrm{x})>0\) for all real numbers \(\mathrm{x}\), then \(\mathrm{f}\) increases without bound. (D) The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Column - I \(\quad\) Column - II (A) If \(x^{2}+y^{2}=1\), then minimum value \(x+y\) is (P) \(-3\) (B) If the maximum value of \(y=a \cos x-\frac{1}{3} \cos 3 x\) occurs (Q) \(-\sqrt{2}\) when \(x=\frac{\pi}{6}\), then the value of \(^{\prime} a\) ' is (C) If \(f(x)=x-2 \sin x, 0 \leq x \leq 2 \pi\) is increasing in the interval (R) 3 \([a \pi, b \pi]\), then \(a+b\) is (D) If equation of the tangent to the curve \(\mathrm{y}=-\mathrm{e}^{-\mathrm{u}^{2}}\) where it (S) 2 crosses the y-axis is \(\frac{x}{p}+\frac{y}{q}=1\), then \(p-q\) is
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